Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle
IINVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES AND THEFERMIONIC CRYSTALLINE EQUIVALENCE PRINCIPLE
ARUN DEBRAY
Abstract.
Freed-Hopkins [FH19a] give a mathematical ansatz for classifying gapped invertible phases ofmatter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slightgeneralization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatialsymmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a“fermionic crystalline equivalence principle,” as predicted in the physics literature. Using this and the Adamsspectral sequence, we compute classifications of some classes of phases with a point group symmetry; in caseswhere these phases have been studied by other methods, our results agree with the literature.
Contents
0. Introduction 11. Phases on a G -space: the general principle 62. The fermionic crystalline equivalence principle 123. Computations in examples: summary of results and some generalities 164. Examples: rotations and reflections 235. Examples: tetrahedral, octahedral, and icosahedral symmetries 416. Glide symmetry protected phases 707. Conclusion and outlook 72References 730. Introduction
The classification of topological phases of matter has been the subject of intensive research in condensed-matter physics and nearby areas of mathematics for the last decade, but difficult problems still remain: forexample, there is not yet an accepted mathematical definition of a topological phase of matter, so researchersmust study these systems using ansatzes or heuristic definitions of phases. Restricting to invertible phases,also known as symmetry-protected topological (SPT) phases , simplifies the classification question, but definingthese phases precisely is also still an open problem. Freed-Hopkins [FH16a] make an ansatz modeling SPTphases using reflection-positive invertible field theories (IFTs), then classify these IFTs using homotopytheory. This approach has been successfully employed in several cases to study examples of SPTs, as in[FH16a, Cam17, WWW18, FHHT20, GOP +
20, PW20].Condensed-matter physicists are also interested in invertible phases in more general settings, includinginvertible phases on a particular space Y , as in [Ran10], or invertible phases symmetric for a group G actingon space, such as phases on the plane which have a rotation symmetry. These spatial symmetries are oftenpresent in real-world examples of topological phases of matter (see [WACB16, MYL +
17] for one example),and can be modeled by lattice Hamiltonian systems in which the symmetry group also acts on the lattice,though again providing precise definitions is still open. In the case where G is a crystallographic group actingon Y = R d , these systems are called crystalline SPT phases . Freed-Hopkins’ field-theoretic approach does notdirectly generalize to this setting, but there is a general ansatz of Kitaev [Kit13, Kit15] that groups of phaseson Y for a fixed symmetry type should define a generalized homology theory. Freed-Hopkins [FH19a] applythis to propose a classification of invertible phases in the presence of a G -action on space using equivariantgeneralized homology. Date : February 8, 2021. a r X i v : . [ m a t h - ph ] F e b ARUN DEBRAY
Researchers interested in computing groups of crystalline SPT phases provide crystalline equivalenceprinciples , including the first such proposal of Thorngren-Else [TE18] and subsequent work in [JR17, CW18,FH19a, ZWY +
20, ZYQG20]. Crystalline equivalence principles are arguments that groups of crystallineSPT phases are isomorphic to groups of ordinary SPT phases, where the symmetry type is modified. Thetheory is well-understood for symmetry types such as O n and SO n , corresponding to the physicists’ notion ofbosonic SPT, but for fermionic SPTs, corresponding to symmetry types such as Spin n , Spin cn , Pin ± n , etc., thestory is more complicated. Cheng-Wang [CW18], Zhang-Wang-Yang-Qi-Gu [ZWY + G -space Y in which thesymmetry type can be merely locally constant over space and can mix with G , including as a special casespatial symmetries mixing with fermion parity. Given data L expressing this mixing and variance of thesymmetry type, we define phase homology groups of Y , denoted Ph G ∗ ( Y, L ), and our ansatz predicts that thegroup of such invertible phases is isomorphic to Ph G ( Y, L ). Providing this ansatz is an additional goal of thispaper, and is necessary input for our FCEP: the ansatz reexpresses the FCEP as an isomorphism betweencertain phase homology groups and groups of IFTs, as we state and prove in Theorem 2.8. This is the firsthomotopy-theoretic account of an FCEP, and to the best of our knowledge is the first fully general version ofthe FCEP in the literature.As a corollary of the FCEP, the computation of phase homology groups that represent groups of point-group-equivariant fermionic phases reduces to computations of bordism groups; this paper’s third goal isto make these computations in several examples, both for the purpose of testing our ansatz by comparingit to established predictions in physics, and for making additional predictions of groups of crystalline SPTphases in as yet unstudied settings. For symmetry types that have been studied before by other methods, ourcomputations agree with the literature, bolstering our ansatz.Now we go into a little more detail about these ansatzes and theorems. Freed-Hopkins [FH19a] formulatean ansatz for invertible phases of matter on a topological space Y equipped with an action of a compactLie group G . First, specify the symmetry type of the theory as a map ρ : H → O, where O := lim −→ n O n is theinfinite orthogonal group and H is a topological group. From this data we can form a Madsen-Tillmannspectrum MTH , whose homotopy groups compute the bordism groups of manifolds with an H -structure onthe tangent bundle. Let I Z denote the Anderson dual of the sphere spectrum and E := Map( MTH , Σ I Z ). Ansatz 0.1 (Freed-Hopkins [FH19a, Ansatz 3.3]) . The abelian group of isomorphism classes of phases on Y equivariant for a G -symmetry that does not mix with the symmetry type H is the Borel-equivariantBorel-Moore homology group E hG , BM ( Y ).We will define equivariant Borel-Moore homology in the generality we need in Definition 1.17.When G is trivial and Y = R n , the group of phases in Ansatz 0.1 is naturally isomorphic to [ MTH , Σ d +2 I Z ],which Freed-Hopkins [FH16a] show is the classification of invertible field theories with symmetry type H . When Y = R d and G is a crystallographic group, this group of phases is expected to model the classificationof crystalline SPT phases with this symmetry type, and indeed, Freed-Hopkins [FH19a, Example 3.5] prove aversion of the bosonic crystalline equivalence principle of Thorngren-Else [TE18] as a consequence of theiransatz, matching physicists’ predictions.For fermionic phases, Ansatz 0.1 is not the full answer, and providing the full answer is a major goal ofthis paper. Physicists distinguish between phases with “spinless fermions” and “spin-1 / G mixes with fermion parity. For example, one could consider phases onthe plane equivariant for a C rotation symmetry, and either ask that fermions’ spin is unaffected by thespatial rotations, or that a full spatial rotation flips the spin on the fermion. This is reminiscent of thebetter-understood dichotomy of fermionic phases with a time-reversal symmetry T : one may have T = 1 or This result is conditioned on a conjecture about non-topological invertible theories; at present, we have as a theorem onlythat the invertible TFTs are classified by the torsion subgroup of this group. This is discussed by Freed-Hopkins [FH16a, §5.4]and Freed [Fre19, Lecture 9].
NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 3 T equal to the fermion parity operator. These two classes of phases are modeled with different symmetrytypes, and similarly we use different data to model crystalline phases with spinless vs. spin-1 / H and the spatial symmetry group G ,we generalize Freed-Hopkins’ setup slightly using parametrized homotopy theory, considering local systems f of symmetry types over the base Y . These give rise to local systems of Thom spectra; if Y has a G -action weobtain from f a local system L of Borel-equivariant Thom spectra, modeled as a functor from Y , though ofas an ∞ -groupoid, to the ∞ -category S p G of Borel-equivariant spectra. Let L := Map(– , Σ I Z ) ◦ L as maps Y → S p G , where I Z has trivial G -action. We define the equivariant phase homology Ph G ∗ ( Y ; L ) to be theequivariant Borel-Moore homology of the local system L : Y → S p G . Ansatz 1.22.
The group of G -equivariant invertible phases on Y for this data is isomorphic to the equivariantphase homology group Ph G ( Y ; f ).When f is trivializable, this reduces to Ansatz 0.1; in general, though, it allows the symmetry type to mixwith the spatial symmetry, or to be merely locally constant on Y .Now we specialize to the cases of spinless and spin-1 / G and H do notmix, so we use the data of a constant local system of symmetry types and recover Freed-Hopkins’ originalansatz. For spin-1 / G by H (0.2) 0 (cid:47) (cid:47) H (cid:47) (cid:47) e H (cid:47) (cid:47) G (cid:47) (cid:47) , together with a representation λ : G → O d dictating how G acts on space. In the cases we consider in thispaper, H = Spin or H = Spin c , and we specify e H by way of the central extension(0.3) 0 (cid:47) (cid:47) µ (cid:47) (cid:47) e G (cid:47) (cid:47) G (cid:47) (cid:47) w ( V λ ) + w ( V λ ) ∈ H ( BG ; µ ), where V λ → BG is the associatedvector bundle to the representation λ and µ is the group of square roots of unity. Then, e H := H × µ e G .Using this data, we build an equivariant local system f of symmetry types, obtaining a phase homologygroup Ph G ( R d , f ) that we predict is isomorphic to the group of invertible phases for this data.The FCEP, previously studied in special cases by [CW18, TE18, ZWY +
20, ZYQG20], identifies groups ofcrystalline SPT phases with groups of fermionic SPT phases with an internal G -symmetry — but exchangingsymmetry types: spinless crystalline phases correspond to spin-1 / G -symmetry using IFTs, and followingFreed-Hopkins [FH16a] and the excellent overview by Beaudry-Campbell [BC18], these groups of TFTs canbe expressed in terms of bordism groups of certain Thom spectra. Standard techniques in algebraic topology,notably the Adams spectral sequence over A (1), can be used to compute these bordism groups, so oneapplication of a general version of the FCEP is to provide access to tractable tools for computing groups ofcrystalline SPT phases.One of the major aims of this paper is to state and prove as a theorem a version of the FCEP, identifyingphase homology groups with groups of IFTs; then Ansatz 1.22 translates this into a statement about crystallineSPTs and ordinary SPTs. In Definitions 2.3 and 2.4, we define the symmetry types for spinless and spin-1 / G -symmetry. In general these definitions are a little technical, but when thespatial representation λ factors through SO d ⊂ O d , the spinless internal symmetry type is H × G → O andthe spin-1 / H × µ e G → O, with the maps induced by the projection onto the first factor.
Theorem 2.8 (Fermionic crystalline equivalence principle) . Fixing data of G , H , λ , etc. as above, let f , f / denote the local systems of symmetry types for the case of spinless, resp. spin- / fermions. Then Ph G ( R d ; f ) is isomorphic to the group of deformation classes of d -dimensional IFTs for the spin- / internal symmetrytype, and Ph G ( R d , f / ) is isomorphic to the group of deformation classes of d -dimensional IFTs for thespinless internal symmetry type. The proof has two key steps.(1) Phase homology groups are defined using equivariant parametrized homotopy theory. Proposi-tion 1.32 reexpresses them using ordinary homotopy theory, as homotopy groups of a Thom spectrum We also specify some additional data; see Data 2.1 in §2 for the full details.
ARUN DEBRAY built from a virtual vector bundle over B e H . The proof uses the Ando-Blumberg-Gepner-Hopkins-Rezk [ABG + + e H n → H n + d × G and showing that it induces a homotopy equivalence on Thom spectra, implying that phase homologygroups are determined by H -bordism groups of a Thom spectrum over BG . Our proof is modeled ona fairly general shearing theorem in Freed-Hopkins [FH16a, §10].After these two steps, the proof of Theorem 2.8 amounts to looking at the Thom spectra for the internalsymmetry types and noticing that we end up with equivalent Thom spectra over BG in the cases we want toequate.With this tool in hand, we can compute phase homology groups for point groups acting on R d , which areour model for groups of fermionic phases equivariant for point group symmetries. We do these computationsfor many 2d and 3d point groups, for both spinless and spin-1 / H = Spin, resp. Spin c ). Our computations use two avatars of the Adams spectralsequence. It is well-known that low-dimensional spin bordism can be computed using connective ko -homologyand the Adams spectral sequence over A (1), and there is an excellent introduction to this technique byBeaudry-Campbell [BC18], but we also use a variant, computing spin c bordism via ku -homology and theAdams spectral sequence over E (1), e.g. in §4.4.3. This is hardly a new idea, but there appear to be noexamples of this specific kind of computation in the literature before now. We hope that our computationsserve as useful examples of how to use this version of the Adams spectral sequence for spin c bordism; thiscould be of independent interest.For 2d point groups, these phases have been studied in the physics literature using very different methods.We compare our results with those of other researchers in §4.1.4, §4.2.4, §4.3.4, and §4.4.5, and find agreement,providing evidence in favor of Freed-Hopkins’ ansatz and our generalization. However, there is not yet workon fermionic crystalline SPT phases for most 3d point groups, so our computations are predictions. We domany computations and make many predictions, and in §3.1 we collect a few that we think are relativelyinteresting or accessible. For example: Theorem.
Let A act on R as the orientation-preserving symmetries of a tetrahedron. Then Ph A ( R , f ) vanishes, where f is the local system of symmetry types for either spinless or spin- / fermions in bothAltland-Zirnbauer classes D and A. This is a combination of Theorems 5.4, 5.6, and 5.8. Therefore, assuming Ansatz 1.22, there are nonontrivial spinless nor spin-1 / TP d ( H ) denotes thegroup of d -dimensional SPT phases with symmetry type H , then the group of d -dimensional glide SPTs isisomorphic to TP d − ( H ) ⊗ Z /
2. Xiong-Alexandradinata [XA18] derive this classification using physics-basedarguments. We use Freed-Hopkins’ ansatzes [FH16a, FH19a] to translate Lu-Shi-Lu’s conjecture into astatement about phase homology groups and prove it.Recall E := Map( MTH , Σ I Z ) and let c Ph Z ∗ ( R d , E ) denote the kernel of the forgetful map from Z -equivariantphase homology to nonequivariant phase homology, where Z acts on R d by glide translations, and E → R d isthe constant local system. This kernel models Lu-Shi-Lu’s group of glide SPTs, as they require glide SPTs tobe trivial in the absence of the glide symmetry. Theorem 6.4.
There is a natural isomorphism c Ph Z ( R d ; E ) ∼ = E − ( d − ⊗ Z / . This provides additional evidence in favor of the ansatz.We want to mention that there are other homotopy-theoretic approaches to the study of phases of matterwith a spatial symmetry, including those of Antolín Camarena, Sheinbaum, and collaborators [AACSS16, SC20]and Cornfeld-Carmeli [CC21]. These authors deal with free fermion phases, which are out of scope of thispaper, though see §7.1.0.1.
Reader’s guide to the different sections.
Overview:
NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 5 • In §§1–2 we discuss general aspects of our model for phases on a G -space Y and prove the FCEP.These sections involve the most homotopy theory. • In §§3–5 we make phase homology calculations which according to Ansatz 1.22 calculate groups offermionic crystalline SPT phases for which the symmetry group is a point group. We collect theresults of these computations in Tables 1, 2, 3, 4, and 5, and summarize the methods of computationin §3.2. • In §6 we consider phases on R d with a glide symmetry, and prove a theorem computing the corre-sponding phase homology classification.Now a little more detail. In §1, we use Borel-equivariant parametrized homotopy theory to state a mildgeneralization of Freed-Hopkins’ ansatz on invertible phases with spatial symmetry. In §1.1, we considerphases on a space Y without a group action, using local systems of symmetry types (Definition 1.3). Wedefine phase homology and in Ansatz 1.10 express the group of invertible phases for such a local systemin terms of phase homology. This is a slight generalization of [FH19a, Ansatz 2.1]. In §1.2, we allowgroup actions, defining equivariant local systems of symmetry types and equivariant Borel-Moore homologyfor a local system for the purpose of formulating Ansatz 1.22 expressing groups of invertible phases for aspatial symmetry in terms of equivariant phase homology. This is a minor generalization of Freed-Hopkins’ansatz [FH19a, Ansatz 3.3] to the parametrized setting. Then, in §1.3, we specialize to the case relevant tothe FCEP, defining the local systems of symmetry types for spatial symmetries that mix with fermion parity.We prove Proposition 1.32 expressing the phase homology groups for this data in terms of nonequivariant,nonparametrized homotopy theory, and do not need equivariant or parametrized homotopy theory in the restof the paper.Next, §2, whose goal is to state and prove the FCEP. We begin in Definitions 2.2, 2.3, and 2.4 by definingthe spinless and spin-1 / G acting on space)and internal ( G not acting on space) symmetries, and use these definitions to state our FCEP theorem inTheorem 2.8, identifying phase homology groups for these local systems in terms of groups of IFTs. Asmentioned, the nontrivial part of the proof runs a shearing argument to simplify a Thom spectrum over B e H into a smash product of MTSpin and a Thom spectrum over BG . In §2.1, we prove Theorem 2.11accomplishing this in class D, for which H = Spin. Then, in §2.2, we prove Theorem 2.24, which is theanalogous theorem in class A, i.e. for H = Spin c , via a similar proof. Finally, in §2.3, we combine thesearguments to prove Theorem 2.8.In §3, we address a few generalities related to the FCEP before studying it in examples. First, in §3.1,we provide a summary of some phases or phenomena newly predicted by our computations which might beinteresting to investigate further. In §3.2, we introduce and review the tools from algebraic topology we needto make these computations: the Adams and Atiyah-Hirzebruch spectral sequences. In §3.3, we discuss howto use the Adams filtration to detect when an invertible TFT of e H -manifolds only depends on the underlyingSO × G -structure, which is believed to correspond to detecting which fermionic phases are really bosonicphases that are fermionic in a trivial way. Finally, in §3.4, we state and prove several lemmas needed in thecomputations in the next sections.Then, in §§4–5, we implement this in examples, computing phase homology groups of R d equivariantfor two- and three-dimensional point-group symmetries, which in Ansatz 1.22 are interpreted as groupsof point group equivariant fermionic phases on R d . In all cases we consider Altland-Zirnbauer classes Dand A (corresponding to symmetry types spin and spin c , respectively), and consider phases with spinlessfermions and spin-1 / c bordism groupsof Thom spectra of vector bundles over BG , where G is the point group of interest; we use the Adams andAtiyah-Hirzebruch spectral sequences to determine these bordism groups.In §4, we consider Z / C n acting byrotations (§4.3) and D n acting by rotations and reflections (§4.4). The results of these computations can befound in Tables 1, 2, 3, and 4. Most of these symmetry types have been studied in the physics literature, andwe compare our results with other researchers’.In §5, we study many 3d point groups, including chiral tetrahedral symmetry (§5.1), pyritohedral symmetry(§5.2), full tetrahedral symmetry (§5.3), chiral octahedral symmetry (§5.4), full octahedral symmetry (§5.5),chiral icosahedral symmetry (§5.6), and full icosahedral symmetry (§5.7). In all cases, we study phases withspinless and spin-1 / ARUN DEBRAY
In §6, we discuss phases equivariant for a glide reflection symmetry. Lu-Shi-Lu [LSL17] conjecture ageneral classification of such phases, and we translate their conjecture into a statement on phase homologygroups using Freed-Hopkins’ ansatz, then prove that statement. Finally, in §7, we suggest some directions forfurther research.
Acknowledgments.
I gratefully thank my advisor, Dan Freed, for his constant help, guidance, and patience.In addition, this paper benefited from conversations with Dexter Chua, Miguel Montero, Riccardo Pedrotti,Daniel Sheinbaum, Luuk Stehouwer, Ryan Thorngren, Juven Wang, and Qing-Rui Wang. Thank you to all.A portion of this work was supported by the National Science Foundation under Grant No. 1440140 whilethe author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, duringJanuary–March 2020. 1.
Phases on a G -space: the general principle We reprise the ansatz of Freed-Hopkins [FH19a, Ansatzes 2.1, 3.3] on invertible phases on a G -space,though we need to generalize it: physicists often consider crystalline phases in which the symmetry acting onspacetime mixes with the internal symmetry (e.g. a reflection squaring to ( − F ), leading us to generalizefrom homology to twisted homology.What we do not do is define a phase of matter. Precisely defining topological phases of matter, even inthe absence of spatial symmetries, is a difficult open question. Our ansatz is a heuristic that these objectscan be classified with what we call phase homology , which we do define.1.1. Invertible phases on a space.
Let Y be a locally compact topological space and C an ∞ -category. Following Ando-Blumberg-Gepner [ABG10, ABG18], we say a C -valued local system on Y is a functor L : π ≤∞ Y → C here π ≤∞ Y is the fundamental ∞ -groupoid of Y . If L : Y → S p is a local system ofspectra, the homology of Y valued in L is L ∗ ( Y ) := π ∗ (hocolim L ), and the cohomology of Y valued in L is L ∗ ( Y ) := π ∗ (holim L ); this generalizes (co)homology with local coefficients.Given a subspace j : Y , → Y , we also define relative homology groups: j induces a map j ∗ : hocolim Y L| Y → hocolim Y L , and we define L ( Y, Y ) := π ∗ (cofib( j ∗ )). Relative cohomology is analogous. Definition 1.1.
Assume that the one-point compactification Y of Y is a finite CW complex and L extendsto a local system L : Y → S p . Choose such an extension L over the basepoint ∗ . The Borel-Moore homology of Y valued in L is(1.2) L BM , ∗ ( Y ) := L ∗ ( Y , ∗ ) . Definition 1.1 appears to depend on the choice of extension of L to Y , but given two choices of extension, thecofibers of the induced maps hocolim L| ∗ → hocolim L are equivalent, hence compute the same Borel-Moorehomology groups.When L is constant, this recovers the usual notion of Borel-Moore (generalized) homology [BM60, Mil95].Recall that a symmetry type is a space B with a map f : B → B O. Definition 1.3. A local system of symmetry types over the space Y is a local system on Y valued in the ∞ -category of spaces with a map to B O.This is closely related to Raptis-Steimle’s definition of parametrized tangential structures [RS17, §2].Symmetry types often arise as the stabilizations in n of maps Bρ n : BH n → B O n induced from representa-tions ρ n : H n → O n ; see [FH16a, §2] for a general discussion. Likewise, the local systems of symmetry typeswe consider arise from BH -bundles over Y .We repeatedly use the notion of Thom spectra ; the definition given by Freed-Hopkins [FH16a, §6.1.4]covers the cases we need. There are different definitions of ∞ -categories; we work with quasicategories as developed by Joyal [Joy02] and Lurie [Lur09],so as to follow [ABG10, ABG18]. However, this paper does not depend on implementation-specific details. See [ABG18, §2] formore information and some useful references. This is not the only approach to parametrized homotopy theory; see also May-Sigurdsson [MS06] and Braunack-Mayer [BM19]. Thom spectra have been heavily studied in homotopy theory; key references include Thom [Tho54], Atiyah [Ati61],May-Quinn-Ray-Tornehave [May77], and Ando-Blumberg-Gepner-Hopkins-Rezk [ABG + + NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 7
Definition 1.4.
Given a representation ρ n : H n → O n or ρ : H → O, where O := lim −→ n O n , we introducenotation for several Thom spectra. Let V n → B O n and V → B O denote the tautological vector bundle, resp.the tautological stable vector bundle. By convention, V → B O has rank zero.(1) The Thom spectra MH n , resp. MH , are the Thom spectra of ( Bρ n ) ∗ V n → BH n , resp. ( Bρ ) ∗ V → BH .(2) The Madsen-Tillmann spectra [MT01, MW07]
MTH n , resp. MTH , are the Thom spectra of ( Bρ n ) ∗ ( − V ) → BH n , resp. ( Bρ ) ∗ ( − V ) → BH .We will use H ∈ { O , SO , Spin , Spin c , Pin ± , Pin c } ; in all of these cases, ρ is the usual map H → O used in,e.g., [FH16a].
Remark . Some Thom spectra go by many names. The notation RP ∞ n denotes ( B O ) nV , and similarly CP ∞ n := ( B SO ) nV . Thus, for example, Σ MTSO , Σ MTU , and Σ CP ∞− all refer to ( B SO ) − V . Definition 1.6.
The
Anderson dual of the sphere spectrum [And69, Yos75] is a spectrum I Z satisfying theuniversal property that for any spectrum X , there is a natural short exact sequence(1.7) 0 (cid:47) (cid:47) Ext( π n − ( X ) , Z ) (cid:47) (cid:47) [ X, Σ n I Z ] (cid:47) (cid:47) Hom( π n ( X ) , Z ) (cid:47) (cid:47) . As all such spectra are equivalent, we refer to “the” Anderson dual of the sphere spectrum to mean anyparticular choice of I Z .(1.7) splits, but not naturally, implying a non-natural isomorphism from [ X, Σ n I Z ] to the direct sum of thetorsion summand of π n − ( X ) and the free summand of π n ( X ). We often use this fact implicitly, calculating π ∗ ( X ) but depending on the reader to rearrange it into [ X, Σ ∗ I Z ]. For more on I Z and its appearance in thiscontext, see Freed-Hopkins [FH16a, §5.3, §5.4].Let Th : T op /B O → S p denote the Thom spectrum functor and I : S p op → S p denote the functorMap(– , Σ I Z ). Definition 1.8.
Let Y be a locally compact space and f : Y → T op /B O be a parametrized symmetry typeon Y . The phase homology of this data, denoted Ph ∗ ( Y ; f ), is the Borel-Moore homology(1.9) Ph ∗ ( Y ; f ) := ( I ◦ Th ◦ f ) BM , ∗ ( Y ) . Ansatz 1.10.
With Y and f as in Definition 1.8, the group of invertible topological phases on Y for thelocal system of symmetry types f is the phase homology group Ph ( Y ; f ).Again, this is not a mathematical definition, but rather a heuristic. Remark . When f is constant, Ansatz 1.10 is the original ansatz of Freed-Hopkins [FH19a, Ansatz 2.1]. Inthat case, the ansatz builds on the idea that invertible phases on Y are related to families of reflection-positiveinvertible field theories on Y . The generalization to nonconstant f allows one to prescribe how the symmetrytype of the family varies along Y . For example, one might want to consider families of phases in which themonodromy around a loop in Y acts by orientation reversal.1.2. Invertible phases on a G -space. Our model for invertible crystalline phases requires considering thecase where a compact Lie group G acts on Y . Again we closely follow Freed-Hopkins [FH19a, §3] but usingtwisted Borel-Moore homology.Throughout this section, G is a Lie group; unlike in [FH16a, FH19a], we do not need G to be compact.Indeed, in the study of crystalline phases, G is often an infinite discrete subgroup of Isom( E n ), and wewill consider one such example in §6. We work with the ∞ -category S p G of Borel G -equivariant spectra ,whose objects can be modeled by data of a sequence of G -spaces X n together with G -equivariant mapsΣ X n → X n +1 . Notions of homotopy equivalence, etc., are as in [FH16a, 6.1], and do not require theircompactness assumption on G . There are a few different notions of G -spectra in the equivariant homotopy theory literature, and their names can beconfusing. Borel G -equivariant spectra can be thought of as “spectra with a G -action” or “spectra living over BG ,” and aredifferent from genuine G -spectra , which have a richer structure. To a geometer, “equivariant (generalized) cohomology” usuallymeans the Borel theory, but to a homotopy theorist, it means the genuine theory. See [Sul20, §2.1] for a detailed introductioninto the different names and notions of G -spaces and G -spectra. ARUN DEBRAY
Definition 1.12.
Suppose G admits a finite-dimensional, real orthogonal representation λ : G → O d . Theone-point compactification of R d with this G -action is a G -space denoted S λ and called a representationsphere .The suspension functor Σ λ := S λ ∧ – is not invertible in G -spaces, but upon stabilization is invertiblein Borel G -spectra; we denote its inverse by Σ − λ . Given a virtual G -representation V = λ − µ (i.e. aformal difference of two finite-dimensional real orthogonal representations), we define the Borel G -spectrum S V := Σ − µ Σ ∞ S λ . We will let S denote the sphere spectrum with trivial G -action. Definition 1.13.
Let Y be a G -space and L : Y → S p G be a local system. The (Borel-)equivariant homology of Y with respect to L is denoted L G ∗ ( Y ) and defined to be(1.14) L G ∗ ( Y ) := π ∗ (Map S p G ( S , hocolim L ) hG ) , where (–) hG : S p G → S p denotes the homotopy fixed-points functor.If j : Y , → Y is an inclusion of G -spaces, it induces a map(1.15) j ∗ : Map S p G ( S , hocolim Y L| Y ) hG −→ Map S p G ( S , hocolim Y L ) hG , and we define the relative (Borel-)equivariant homology (1.16) L G ∗ ( Y, Y ) := π ∗ (cofib( j ∗ ))as in the nonequivariant case. Definition 1.17.
Let Y be a G -space and L : Y → S p G be an S p G -valued local system. Assume that theone-point compactification Y of Y is a CW complex and L extends to a local system L : Y → S p G . Choosesuch an extension L . The equivariant Borel-Moore homology of Y valued in L is(1.18) L G BM , ∗ ( Y ) := L G ∗ ( Y , ∗ ) . Just like Definition 1.1, this does not actually depend on the choice of extension.
Definition 1.19.
Let Y be a G -space. A G -equivariant local system of symmetry types is a G -space B anda G -equivariant map f : B → Y × B O, where B O has a trivial G -action.Taking the Thom spectrum of the map to B O defines a local system Th ◦ f : Y → S p G . Definition 1.20.
Let Y be a G -space whose one-point compactification is a finite CW complex, and let f : B → Y × B O be a G -equivariant local system of symmetry types for Y . The G -equivariant phase homology of this data, denoted Ph G ∗ ( Y ; f ), is the equivariant Borel-Moore homology(1.21) Ph G ∗ ( Y ; f ) := ( I ◦ Th ◦ f ) G BM , ( Y ) . Ansatz 1.22.
With Y and f as in Definition 1.20, the group of invertible topological phases on Y for theequivariant local system of symmetry types f is the G -equivariant phase homology group Ph G ( Y ; f ).Again, this is a heuristic and not a definition. When G is a discrete subgroup of Isom( E n ) (e.g. a wallpaperor space group) acting on Y = E n , these phases are called crystalline SPT phases in the physics literature.1.3. Mixing internal and crystalline symmetries.
The fermionic crystalline equivalence principle isabout invertible topological phases in which an internal symmetry mixes with the symmetry group acting onspace. In this section, we construct the equivariant local systems of symmetry types for these phases. First,we review how mixing of symmetries is handled in the purely internal case in Example 1.23; then we addressthe case of spatial symmetries in Proposition 1.32, showing how to reduce the computation of the relevantequivariant phase homology groups to a nonparametrized question. We will simplify these computationsfurther in §2 when we discuss the FCEP in more detail, then study several examples in §§4–5.
Example 1.23 (Mixing for internal symmetries) . In the study of SPTs, one commonly encounters symmetrytypes where there are two different symmetries present, such as time reversal and fermion parity, but theymix, meaning the group they generate is not a product of the individual symmetry groups, but rather anextension. For example, we could ask for a generator T of the group of time-reversal symmetries to square tothe fermion parity ( − F , via the extension 0 → Z / → Z / → Z / →
0, rather than considering phaseswhere T = 1, corresponding to the split extension 0 → Z / → Z / × Z / → Z / → NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 9
Freed-Hopkins [FH16a] make the ansatz that SPT phases are classified up to equivalence by their low-energylimits, which are invertible field theories. The symmetry type is expressed as an H n -structure, where H n is agroup with a map to O n ; mixing manifests as an extension involving the base symmetry type (e.g. Spin n for fermionic phases) and the additional symmetry. For example, the two cases of time-reversal symmetrysquaring to the identity or to fermion parity are represented by the extensions1 (cid:47) (cid:47) Spin n (cid:47) (cid:47) Pin + n (cid:47) (cid:47) Z / (cid:47) (cid:47) (cid:47) (cid:47) Spin n (cid:47) (cid:47) Pin − n (cid:47) (cid:47) Z / (cid:47) (cid:47) , (1.24b)respectively, together with the standard maps Pin ± n → O n .When one of the groups we want to mix acts on space, we can specify a mixed symmetry type by thefollowing data: • a symmetry type ρ n : H n → O n , called the base symmetry type , • the point group symmetry λ : G → O d , • an extension(1.25) 1 (cid:47) (cid:47) H n (cid:47) (cid:47) e H n (cid:47) (cid:47) G (cid:47) (cid:47) • an extension e ρ n : e H n → O n of ρ n : H n → O n .Freed-Hopkins [FH16a, §9.2] relate Altland-Zirnbauer’s symmetry classes of condensed-matter systems [Zir96,AZ97] to ten symmetry types in topology. Using this, we call the case H = Spin the class D case and H = Spin c the class A case .Let Y be a G -space. Then the map(1.26) Y × E e H n /H n −→ Y is a G -equivariant fiber bundle with fiber BH n , and the total space maps to B O n as specified by the virtualvector bundle(1.27) f : − ( Y × ( E e H n × H n R n )) −→ Y × E e H n /H n . After stabilizing (i.e. letting n → ∞ ), this is an equivariant local system of symmetry types over Y , so hasequivariant phase homology groups Ph G ∗ ( Y ; f ). Under Ansatz 1.22, Ph G ( Y ; f ) models the group of invertibletopological phases on Y in which fermion parity mixes with the spatial symmetry as specified by (1.25).The notion of G -equivariant phases for this symmetry type (without a reference space Y ) is taken to mean G -equivariant phases on R d , where G acts on R d through λ . Remark . We would like to be able to move information between instances ofthis data: for example, there should be forgetful maps from equivariant phases on a space to nonequivariantones, and we model them with maps between phase homology groups for the two local systems of symmetrytypes.Suppose we are given two instances of the data above. That is, we ask for a commutative diagram of Liegroups(1.29) 1 (cid:47) (cid:47) H n (cid:47) (cid:47) ϕ (cid:15) (cid:15) e H n (cid:47) (cid:47) e ϕ (cid:15) (cid:15) G (cid:47) (cid:47) ϕ G (cid:15) (cid:15) (cid:47) (cid:47) H n (cid:47) (cid:47) e H n (cid:47) (cid:47) G (cid:47) (cid:47) ρ n : H n → O n and ρ n : H n → O n , λ : G → O d and λ : G → O d , and e ρ n : e H n → O n and e ρ n : e H n → O n which commute with the vertical maps in (1.29). Fix a G -space Y ; then through (1.27) thisdefines equivariant local systems of symmetry types f for G , resp. f for G . The maps between the datainduce a pullback or forgetful map ϕ ∗ : Ph G ∗ ( Y ; f ) → Ph G ∗ ( Y ; f ), where G acts on Y through ϕ G . Using This “tenfold way” is a relativistic version of Dyson’s threefold way [Dys62], and appears in many contexts in physics,including [Kit09, RSFL10, FM13, WS14, FH16a, KZ16, GM20, IT20].
Ansatz 1.22, we interpret this pullback map realizing an invertible phase on Y with a G -symmetry to aphase with a G -symmetry.The construction of ϕ ∗ amounts to checking that diagrams you would expect to commute do in factcommute. The data we gave induces a commutative diagram(1.30) − ( Y × ( E e H n × H n R n )) (cid:47) (cid:47) (cid:15) (cid:15) Y × E e H n /H n (cid:15) (cid:15) − ( Y × ( E e H n × H n R n )) (cid:47) (cid:47) Y × E e H n /H n . The rows define equivariant local systems symmetry types; then f and f are the maps to Y × B O. Let ϕ ◦ : S p G → S p G be the map in which G acts on Borel G -spectra through ϕ ; then, upon applying I ◦
Th, weobtain local systems L , resp. L of Borel G -, resp. G -spectra. To define phase homology, we assumed that anextension L of L to Y exists, so choose such an extension; then L := L ◦ ϕ ◦ is an extension of L . We obtainfrom the inclusion ∗ , → Y a commutative diagram of spectra(1.31) Map S p G ( S , hocolim ∗ L | ∗ ) hG (cid:47) (cid:47) (cid:15) (cid:15) Map S p G ( S , hocolim ∗ L| ∗ ) hG (cid:15) (cid:15) Map S p G ( S , hocolim Y L ) hG (cid:47) (cid:47) Map S p G ( S , hocolim Y L ) hG . Thus, we get a map between the cofibers of the vertical arrows, and π ∗ of that map is the desired map onphase homology.For us there are two particularly important examples.(1) Let H n = H n and G = 1, which forces e ϕ : e H n → e H n to be the inclusion H n → e H n . The aboveconstruction produces a map from H -equivariant phase homology to nonequivariant phase homologyon Y , which we interpret as modeling the forgetful map from phases with a G -symmetry to phaseswithout a G -symmetry.(2) Let G = G , H n = SO n . and H n be either Spin n or Spin cn , with ϕ the usual map. In this casethe pullback map goes from equivariant phase homology where the base symmetry type is SO toequivariant phase homology where the base symmetry type is Spin or Spin c . We interpret this inphysics as modeling the procedure that regards a bosonic phase as a fermionic phase by adding sometrivial fermionic degrees of freedom. This is analogous to the procedure which regards an orientedTFT as a spin TFT that does not depend on the spin structure.Crucially for computations, we can simplify the equivariant phase homology groups for the symmetrytypes in (1.27) into a description not requiring equivariant or parametrized stable homotopy theory. Proposition 1.32.
There is an isomorphism (1.33) Ph G ( R d ; f ) ∼ = −→ [( B e H ) d − λ − e ρ , Σ d +2 I Z ] natural for changing the symmetry type in the sense of Remark 1.28.Proof. We want to compute the twisted equivariant Borel-Moore homology for this equivariant local systemof symmetry types, where Y = R d with G acting through λ . This amounts to the following: one-pointcompactify to a local system over S λ ; take the colimit of the local system and call it E ; then compute [ S , E ] G (in the notation of [FH19a]; this means π (Map( S , E ) hG )). Now, the local system ( I ◦ Th ◦ f ) : S λ → S p G is nonequivariantly the trivial local system with fiber Map( MTH , Σ I Z ), so E ’ S λ ∧ Map(
MTH , Σ I Z ); ingeneral, G can act nontrivially on both S λ and MTH , but always acts trivially on Σ I Z . Therefore we mayfollow [FH19a, (3.6)] and identify(1.34) Map( S , S λ ∧ Map(
MTH , Σ I Z )) ’ Map( S d − λ ∧ MTH , Σ d +2 I Z ) , though the G -action on S d − λ ∧ MTH is not the diagonal action, but rather the induced G -action on theThom spectrum of the G -equivariant virtual bundle ( d − λ − ρ ) → BH (see [FH16a, §6.2.2]). NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 11
Since G acts trivially on Σ d +2 I Z ,(1.35) Map( S d − λ ∧ MTH , Σ d +2 I Z ) hG ’ Map(( S d − λ ∧ MTH ) hG , Σ d +2 I Z ) . It now suffices to show that(1.36) ( S d − λ ∧ MTH ) hG ’ ( B e H ) − e ρ − λ + d . Ando-Blumberg-Gepner-Hopkins-Rezk [ABG + V → X , identified with a map V : X → B O, is the homotopy colimit(1.37) X V ’ hocolim (cid:0) X V (cid:47) (cid:47) B O BJ (cid:47) (cid:47) B GL ( S ) (cid:47) (cid:47) S p (cid:1) , where the notation means to interpret X as, through its fundamental ∞ -groupoid, providing a diagram in the ∞ -category S p of spectra. Here B GL ( S ) is the classifying space of stable spherical fibrations [Sta63, May77]and BJ : B O → B GL ( S ) is a form of the J -homomorphism [Whi42, May77]. Heuristically, (1.37) saysthat the virtual vector bundle V defines a local system of ∧ -invertible spectra, with the fiber at a point x ∈ X given by S V x , and that the Thom spectrum is obtained from an associated bundle construction.See [ABG + + S d − λ ∧ MTH ) hG = hocolim pt /G (cid:16) hocolim (cid:0) BH d − λ − ρ (cid:47) (cid:47) B O BJ (cid:47) (cid:47) B GL ( S ) (cid:47) (cid:47) S p (cid:1)(cid:17) , (1.38a)where G acts on the spectra in the diagram through its action on λ , as well as on BH , as prescribed bythe extension (1.25). This in particular implies the double homotopy colimit above simplifies into a singlehomotopy colimit over a B e H -shaped diagram: ’ hocolim (cid:16) B e H d − λ − e ρ (cid:47) (cid:47) B O BJ (cid:47) (cid:47) B GL ( S ) (cid:47) (cid:47) S p (cid:17) , (1.38b)which by (1.37) is the Thom spectrum for d − λ − e ρ → B e H , proving (1.36). (cid:3) Our next step in §2 is to simplify ( B e H ) d − λ − e ρ . This allows both for a general formulation of the fermioniccrystalline equivalence principle as well as explicit calculations.The following lemma will be helpful for simplifying Thom spectra. Theorem 1.39 (Relative Thom isomorphism) . Let ρ : H → O be a symmetry type with the two-out-of-threeproperty, i.e. an H -structure on any two of E , F , or E ⊕ F induces one on the third. If V, W → X arevirtual vector bundles such that V has an H -structure, then there is an equivalence (1.40) MTH ∧ X W ’ −→ MTH ∧ X V ⊕ W . Proof.
The two-out-of-three property gives
MTH an E ∞ -ring structure, which is needed for some of theconstructions we employ from [ABG + + E → X depends only on the homotopyclass of the map f E : X → B O → B GL ( S ), where the first map is given by E , and the second mapis the J -homomorphism, as in (1.37). Smashing with MTH corresponds to composing f E with the map B GL ( S ) → B GL ( MTH ) induced by the Hurewicz map S → MTH [ABG + MTH ∧ X E only depends on the homotopy type of the map X → B GL ( MTH ).Because
MTH is an E ∞ -ring spectrum, B GL ( MTH ) is a grouplike E ∞ -space, and the composition ψ : B O → B GL ( S ) → B GL ( MTH ) is a map of grouplike E ∞ -spaces, where B O carries the E ∞ structurecoming from direct sum. This means that [ X, B GL ( MTH )] is naturally an abelian group, and that if wedefine classes in this group using virtual vector bundles
V, W → X to map to B O then composing with ψ ,the class of E ⊕ F is the sum of the classes of V and W .An H -structure on V trivializes the map X → B O ψ → B GL ( MTH ) defined by V , so the class of the mapdefined by V ⊕ W is equal to the class of the map defined by W in the abelian group [ X, B GL ( MTH )]. (cid:3) The fermionic crystalline equivalence principle
In this section, our goal is to state and prove the FCEP, Theorem 2.8, identifying phase homology groupsin classes D and A with groups of deformation classes of invertible field theories. Assuming Ansatz 1.22, thisleads to the more familiar version of the FCEP: crystalline equivalence principles are first introduced byThorngren-Else [TE18]: the idea is to equate the classification of crystalline topological phases of matter forsome group G acting on spacetime with a classification of a different kind of topological phases of matter, inwhich G is part of the internal symmetry group. Then one may use preexisting techniques for phases withouta spatial symmetry to classify phases with the specified G -action on space.The best-understood crystalline equivalence principles are for bosonic SPTs, as first considered by Thorngren-Else [TE18]. “Bosonic” does not have a precise mathematical translation here; these are phases for which thesymmetry type is built using SO or O rather than Spin, Spin c , Pin ± , and so on. If a group G acts on space byorientation-preserving symmetries and H is SO or O, the classification of crystalline SPTs in dimension n withsymmetry type H and this G -action is identified with the classification of SPTs for H = SO × G . To whatextent this is an ansatz or a theorem depends on one’s model for crystalline SPTs: Freed-Hopkins [FH19a,Example 3.5] derive it as a corollary of their ansatz. For other derivations of the bosonic crystallineequivalence principle from different ansatzes, see Jiang-Ran [JR17] and Thorngren-Else [TE18, ET19].The fermionic analogue of this statement is more complicated because there are more ways for G tomix with the symmetry type. Thorngren-Else [TE18, §VII.B], Cheng-Wang [CW18], Zhang-Wang-Yang-Qi-Gu [ZWY + G mixeswith H : crystalline phases for which the spatial G -symmetry does not mix with fermion parity correspond tophases with an internal G -symmetry that does mix with fermion parity, and vice versa. Examples of thistwisted correspondence also appear in work of Freed-Hopkins [FH19a, Example 3.5], Guo-Ohmori-Putrov-Wan-Wang [GOP + H = Spin or H = Spin c ), forall compact Lie groups G acting on faithfully on space, and all ways in which G may mix with fermion parity.The slogan “mixed crystalline goes to unmixed internal, and vice versa” is a little hard to glean from theresult when the G -action includes reflections, but we obtain an equivalence from phase homology groupsfor certain equivariant local systems of symmetry types, which under Ansatz 1.22 stands in for groups ofcrystalline SPT phases, to groups of deformation classes of IFTs, which under Freed-Hopkins’ ansatz [FH16a]model groups of phases without spatial symmetries.To precisely state our FCEP, we must fix some data. Data 2.1. • Let H denote the base symmetry type, which today is either of the infinite-dimensional topologicalgroups Spin or Spin c . • Let G be a compact Lie group, λ : G → O d be a faithful representation, and V λ := EG × G R d → BG be the associated vector bundle. • Let ξ : G → O d be another faithful representation and V ξ → BG be the associated vector bundle.Let 1 → µ → e G → G → w ( V λ ) + w ( V λ ) ∈ H ( BG ; µ ).Here µ denotes the group of square roots of unity. • Let e H := H × µ e G . Let ρ be the composition e H → H → O and V → B e H be the associatedtautological vector bundle.For us, ξ and λ are usually the same, but they differ when G = Z / R d by inversion in the case ofspin-1 / ξ is the sign representation σ : Z / → O , but λ = dσ . See §4.2 for more detail. Definition 2.2.
The spin- / equivariant local system of symmetry types for the above data is the G -equivariant parametrized symmetry type f / : B e H → R d × B O which sends x (0 , Bρ ( x )), and in which G acts on R d through λ . The spinless equivariant local system of symmetry types f is defined in the sameway, except using H × G instead of e H . If G acts by reflections, almost as nice of a story is still true, but the internal G -symmetry mixes with H . Thorngren-Else [TE18] and Freed-Hopkins [FH19a, Example 3.5] discuss this case too. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 13
Definition 2.3.
Recall that H is either Spin or Spin c . Let † ∈ {− , c } be − if H = Spin and c otherwise.The spinless internal symmetry type is the symmetry type • ( − V, d − V λ ) : BH × BG → B O, if λ is pin † , or • ( − V, V ξ + Det( V ξ ) − V λ ) : BH × BG → B O, if λ is not pin † .For shorthand, we denote this symmetry type ρ (0) : BH × BG → B O. Definition 2.4.
The spin- / internal symmetry type is the symmetry type(2.5) ( − V, d − V λ ) : BH × BG → B O . For shorthand, we denote this symmetry type ρ (1 /
2) : BH × BG → B O. Remark . The internal symmetry types probably look pretty arbitrary. This is because of the generalityof our setup: in some cases of interest, we can rewrite these symmetry types in ways which more closelyresembles the proposals of Thorngren-Else [TE18, §VII.B], Cheng-Wang [CW18], and Zhang-Wang-Yang-Qi-Gu [ZWY +
20] for the FCEP in specific cases.Suppose λ = ξ and Im( λ ) ⊂ SO d but does not lift across Spin d (cid:16) SO d . Then, the spinless internalsymmetry type simplifies to BH × BG → B O, where the map is just projection onto the first factor followedby the usual map BH → B O. That is, for representations with image contained in SO d , the FCEP switches the“unmixed” (i.e. BH × BG ) and “mixed” (i.e. B ( H × µ e G )) symmetry types when passing between crystallineand internal phases. This matches predictions by Thorngren-Else [TE18] and Cheng-Wang [CW18].Freed-Hopkins [FH16a, Corollary 8.21] show that the group of deformation classes of reflection-positiveIFTs with symmetry type ρ : H → O in (space) dimension n is naturally isomorphic to (2.7) [ MTH , Σ d +2 I Z ] . Theorem 2.8 (Fermionic crystalline equivalence principle) . There are isomorphismsPh Gk ( R d ; f ) ∼ = −→ [ MT ρ (1 / , Σ d + k +2 I Z ](2.9a) Ph Gk ( R d ; f / ) ∼ = −→ [ MT ρ (0) , Σ d + k +2 I Z ] . (2.9b)Assuming Ansatz 1.22, the physics implication of this theorem is that the abelian group of crystalline SPTphases for the spinless equivariant local system of symmetry types is naturally isomorphic to the abelian groupof deformation classes of IFTs for the spin-1 / / We obtain Thom spectra for vector bundles over B e H , and tofinish we must compare these spectra to MTH ∧ ( BG ) E , where E → BG is some rank-zero virtual vectorbundle. This comparison, in the form of shearing arguments , is the core of the proof: we prove Theorem 2.11( H = Spin) and Theorem 2.24 ( H = Spin c ) establishing the homotopy equivalences we need, and afterthat proving Theorem 2.8 amounts to verifying that the outputs of Theorems 2.11 and 2.24 simplifyingthe crystalline symmetry types match the Thom spectra for the internal symmetry types in Definitions 2.3and 2.4.The proofs of Theorems 2.11 and 2.24 resemble the proofs of the more standard equivalences MTPin + ’ MTSpin ∧ ( B Z / − σ (2.10a) MTPin − ’ MTSpin ∧ ( B Z / σ − (2.10b) MTPin c ’ MTSpin c ∧ ( B Z / ± (1 − σ ) (2.10c) MTSpin c ’ MTSpin ∧ ( B SO ) ± (2 − V ) , (2.10d) Strictly speaking, Freed-Hopkins’ theorem classifies only the invertible topological field theories, which form the torsionsubgroup of (2.7), and they conjecture that the entire group classifies all reflection-positive IFTs. For the spinless equivariant symmetry type, this is just [FH19a, Example 3.5]. where σ → B Z / V → B SO denote the respective tautological line bundles. These decompositionswere first proven by Kirby-Taylor [KT90a, Lemma 6] (pin + ), Peterson [Pet68, §7] (pin − ), and Bahri-Gilkey [BG87a, BG87b] (spin c and pin c ). For a unified proof of all of these equivalences, see Freed-Hopkins [FH16a, §10].2.1. Case H = Spin .Theorem 2.11 (Shearing, class D) . Let V → B e H be the tautological bundle.(1) Suppose V ξ admits a pin − structure. Then there is an equivalence (2.12) ( B e H ) d − V λ − V ’ −→ MTSpin ∧ ( BG ) d − V λ . (2) If V ξ does not admit a pin − structure, there is an equivalence (2.13) ( B e H ) d − V λ − V ’ −→ MTSpin ∧ ( BG ) V ξ +Det( V ξ ) − V λ − d − d . We will most often consider case (2) with λ = ξ , in which case we learn ( B e H ) d − λ − V ’ MTSpin ∧ ( BG ) Det( V λ ) − . Proof.
Case (1) is by far the easier of the two: V ξ admits a pin − structure iff w ( V ξ ) + w ( V ξ ) = 0 iffthe extension 1 → µ → e G → G → µ ⊂ e G is central, a splitting induces isomorphisms e G ∼ = µ × G and e H n ∼ = Spin n × G . Passing to classifying spaces, this identifies d − V λ − V : B e H → B O with − V (cid:1) ( d − λ ) : B Spin × BG → B O; then take Thom spectra.On to case (2). In this case, in H ( B e H ; µ ), w ( V ξ )+ w ( V ξ ) = w ( V ), so the map V + V ξ +Det( V ξ ) : B e H → B SO lifts across B Spin → B SO. Choose such a lift.
Proposition 2.14.
The induced map (2.15) ( V + V ξ + Det( V ξ ) , ξ ) : B e H −→ B Spin × BG is a homotopy equivalence commuting with the maps to B SO . The proof is due to Freed-Hopkins [FH16a, §10].
Proof.
We will show that the commutative square(2.16a) B e H (cid:47) (cid:47) B ( π ⊕ π ) (cid:15) (cid:15) B Spin (cid:15) (cid:15) B SO × BG B (id ⊕ ξ ) (cid:47) (cid:47) B SOis homotopy Cartesian. Any two homotopy pullbacks of the same diagram are weakly equivalent, with theweak equivalence intertwining the maps to B SO. Since there is also a homotopy pullback square(2.16b) B Spin × BG (cid:47) (cid:47) (cid:15) (cid:15) B Spin (cid:15) (cid:15) B SO × BG B (id ⊕ ξ ) (cid:47) (cid:47) B SO , then B e H ’ B Spin × BG ; this equivalence is realized by (2.15) because that is the only possibility thatintertwines the maps in (2.16a) and (2.16b).To fulfill the promise that (2.16a) is a homotopy pullback square, begin with the commutative diagram ofshort exact sequences(2.17) 1 (cid:47) (cid:47) µ (cid:47) (cid:47) e H n ( π ,π ) (cid:47) (cid:47) (cid:15) (cid:15) SO n × G (cid:47) (cid:47) id ⊕ ξ (cid:15) (cid:15) (cid:47) (cid:47) µ (cid:47) (cid:47) Spin n + d (cid:47) (cid:47) SO n + d (cid:47) (cid:47) . NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 15
This induces a map of fiber sequences(2.18) B e H B ( π ,π ) (cid:47) (cid:47) (cid:15) (cid:15) B SO × BG B (id ⊕ ξ ) (cid:15) (cid:15) w (cid:47) (cid:47) K ( µ , B Spin (cid:47) (cid:47) B SO w (cid:47) (cid:47) K ( µ , , e.g. B e H is the fiber of w : B SO × BG → K ( µ , (cid:3) Including the maps down to B SO produces the commutative diagram(2.19) B e H ( V + V ξ +Det( V ξ ) ,ξ ) ’ (cid:47) (cid:47) − V (cid:35) (cid:35) B Spin × BG − V + V ξ +Det( V ξ ) (cid:120) (cid:120) B SO . Taking Thom spectra of the vertical maps, the shearing map induces a homotopy equivalence(2.20) ( B e H ) − V ’ −→ MTSpin ∧ ( BG ) V ξ +Det( V ξ ) − d − . To finish, we subtract V λ from the vertical arrows in (2.19), then take Thom spectra again. (cid:3) Case H = Spin c . Let e H n := Spin cn × µ e G , and define e H similarly. The shearing argument is scarcelydifferent than for Theorem 2.11, but it will be useful to rephrase e H n using the circle group T instead of µ .The extension of G by µ defines an extension of G by T by pushing forward along the inclusion µ , → T :(2.21) 1 (cid:47) (cid:47) µ (cid:127) (cid:95) (cid:15) (cid:15) (cid:47) (cid:47) e G (cid:15) (cid:15) (cid:47) (cid:47) G (cid:47) (cid:47) (cid:47) (cid:47) T (cid:47) (cid:47) b G (cid:47) (cid:47) G (cid:47) (cid:47) . In cohomology, this construction is classified by the Bockstein map H ( BG ; µ ) → H ( BG ; Z ). Let b H n :=Spin cn × T b G and b H := Spin c × T b G . The map e G → b G induces maps ϕ n : e H n → b H n and ϕ : e H → b H ; ϕ is thecolimit of the ϕ n s. Lemma 2.22.
The maps ϕ n : e H n → b H n are isomorphisms of Lie groups.Proof. Write down the commutative diagram(2.23) 1 (cid:47) (cid:47) µ (cid:47) (cid:47) e H nϕ (cid:15) (cid:15) (cid:47) (cid:47) SO n × T × G (cid:47) (cid:47) (cid:47) (cid:47) µ (cid:47) (cid:47) b H n (cid:47) (cid:47) SO n × T × G (cid:47) (cid:47) (cid:3) And now we shear. Recall our notation from Data 2.1.
Theorem 2.24 (Shearing, class A) . (1) Suppose V ξ admits a pin c structure. Then there is an equivalence (2.25) ( B b H ) d − V λ − V ’ −→ MTSpin c ∧ ( BG ) d − V λ . (2) If V ξ does not admit a pin c structure, there is an equivalence (2.26) ( B b H ) d − V λ − V ’ −→ MTSpin c ∧ ( BG ) V ξ +Det( V ξ ) − V λ − d +1 − d . Again, we most often use case (2) when λ = ξ , in which case the right-hand side simplifies to MTSpin c ∧ ( BG ) Det( V λ ) − . Proof.
The proof is barely different than that of Theorem 2.11; we indicate only the differences. In thattheorem, the engine of the proof when V ξ was not pin − was the map (2.15) from B (Spin × µ e G ) → B Spin × BG .Here, V ξ is not pin c , so V ξ ⊕ Det( V ξ ) is oriented but not spin c . We have that if β : H ( B b H ; µ ) → H ( B b H ; Z )is the Bockstein, β ( w ( V ξ ) + w ( V ξ ) + w ( V )) = 0, so V + V ξ + Det( V ξ ), interpreted as a map B b H → B SO,lifts to B Spin c . Our analogue of (2.15) is(2.27) ( V + V ξ + Det( V ξ ) , ξ ) : B b H −→ B Spin c × BG.
As in Proposition 2.14, this is a homotopy equivalence commuting with the maps down to B SO. The proof isalmost the same, though we replace Spin with Spin c in (2.16a) and (2.16b), µ with T in (2.17), and K ( µ , K ( Z ,
3) in (2.18). (cid:3)
Putting it together.
The hard work of the proof is already done.
Proof of Theorem 2.8.
By Proposition 1.32,(2.28) Ph G ( R d ; f / ) ∼ = [ X, Σ d +1 I Z ] , where X := ( B e H ) d − V λ − V . Then Theorem 2.11 ( H = Spin) and Theorem 2.11 ( H = Spin c ) split this into MTH ∧ ( BG ) E for some rank-zero virtual vector bundle E . For f , because e H ∼ = H × G , Proposition 1.32gets us to MTH ∧ ( BG ) E without having to shear. The only thing left to do is compare these Thom spectrato Definitions 2.3 and 2.4, and sure enough, they match. (cid:3) Computations in examples: summary of results and some generalities
In the next two sections, we study the fermionic crystalline equivalence principle in many examples wherethe symmetry is given by a two- or three-dimensional point group. Here, we summarize the results and sometakeaways for researchers interested in crystalline phases; for more detailed results of computations of groupsof phases, see Tables 1, 2, 3, 4, and 5.In §3.1, we indicate some example phases predicted by our phase homology calculations that have not beenpreviously studied to our knowledge, and which might have accessible or interesting lattice realizations. Wealso summarize which of our calculations correspond to phases already studied in the literature. In §3.2, webriefly review the computational techniques we use to study phase homology groups, namely the Adams andAtiyah-Hirzebruch spectral sequences. In §3.3, we use the Adams filtration to characterize which invertiblefield theories with e H -structure actually only require weaker structure, such as an SO × G -structure; thisis believed to model the phenomenon in physics of phases which appear to be fermionic, but are in factbosonic phases that are not fermionic in an interesting way. Finally, in §3.4, we gather some lemmas weuse repeatedly in the coming sections. The reader interested in the computations can read §3.1 and §3.2,returning to the other sections later.3.1. Some interesting phases to study.
In §§4–5, we compute equivariant phase homology groups formany 2- and 3-dimensional point groups. Using Ansatz 1.22, these computations yield predictions of groupsof invertible topological phases. This is a lot of data, so we take the opportunity here to highlight which ofour predictions would be interesting to study by other means, e.g. by arguing on the lattice.We first study some cases already present in the literature and find agreement, including reflections inAltland-Zirnbauer classes D and A (§4.1), inversions in classes D and A (§4.2), cyclic groups acting byrotations in classes D and A (§4.3), and dihedral groups acting by rotations and reflections in class D (§4.4).In all cases we consider both spinless and spin-1 / / D n acts by rotations and reflections.(a) In dimension d = 2, we predict using Theorem 4.40 a phase generating a Z / n ≡ Z / n ≡ NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 17 (b) In dimension d = 3, we would be interested in the predicted Z / ⊕ Z / n odd, with eitherspin-1 / Z / n even with spin-1 / Z / ⊕ Z / Z / R for A acting by tetrahedral symmetryand find that for classes A and D and the spinless and spin-1 / Z / n be odd and d be arbitrary. Lemma 4.20 is an argument applied to phase homology whichsuggests that for any symmetry type and choice of mixing, an inclusion Z / ⊂ D n as a reflectioninduces an isomorphism between the group of phases with that symmetry type on R d equivariantfor a reflection, and the group of such phases on R d equivariant for D n acting by rotations andreflections in a plane. Can this be seen another way?Our computations predict plenty of other phases, but many of them either have Adams filtration zero (see§3.3) and therefore are not predicted to be intrinsically fermionic, or have more complicated symmetry types,such a full octahedral symmetry, that would be harder to study on the lattice. Remark . In the computations we make in the next several sections, we generally report more bordismgroups than we need to determine the phase homology groups corresponding to groups of invertible phases:to compute the group of n -dimensional invertible field theories with symmetry type H → O, we need thetorsion subgroup of π n ( MTH ) and the free summand in π n +1 ( MTH ). Bordism has other applications ingeometry and physics, so we usually report all bordism groups π k ( MTH ) that follow from the calculationsthat we need for crystalline phases. When k ≥ n + 1, these provide information about higher-dimensionalcrystalline phases; for k < dim( λ ), though, it is not clear what a crystalline phase could mean when there arenot enough space dimensions for G to act by λ , and we do not give a physical meaning to these computations.See [GOP +
20] for some discussion when space time is dim( λ )-dimensional.3.2. Methods of computation.
In this section, we summarize the techniques we use to make thesecomputations, and gather a few auxiliary lemmas we need along the way. Most of our computations can bereframed as computing certain twisted ko - and ku -homology groups of finite groups in low degrees; the readerinterested in learning how to perform such computations is encouraged to refer to the monographs of Bruner-Greenlees [BG03, BG10] on connective ko - and ku -theory, as well as Beaudry-Campbell’s article [BC18] onusing the Adams spectral sequence to compute ko -theory. Computing spin bordism:
Let ko denote the connective real K -theory spectrum. Anderson-Brown-Peterson [ABP67] show that the Atiyah-Bott-Shapiro map MTSpin → ko [ABS64] is 7-connected,meaning that for any space or spectrum X , the induced map Ω Spin k ( X ) → ko k ( X ) is an isomorphismfor k ≤
7. We often pass between spin bordism and ko -theory without comment. We compute thefree and 2-torsion summands of ko ∗ ( X ) using the Adams spectral sequence; see below. The forgetfulmap MTSpin → MTSO induces an equivalence on odd-primary torsion, so to compute odd-primarytorsion, we typically compute Ω SO ∗ ( X ) via the Atiyah-Hirzebruch spectral sequence, which we alsodiscuss below. Computing spin c bordism: Let ku denote connective complex K -theory. Anderson-Brown-Peterson [ABP67]also produce a 7-connected map MTSpin c → ku ∨ Σ ku ; we will also use the Adams spectral se-quence to determine the free and 2-torsion summands of ku ∗ ( X ), as described below. The forgetfulmap MTSpin c → MTSO ∧ ( B U ) + induces an equivalence on odd-primary torsion, so we computeΩ SO ∗ ( X × B U ), typically with the Atiyah-Hirzebruch spectral sequence.Now we briefly introduce the Adams and Atiyah-Hirzebruch spectral sequences in the ways that we use them. The Adams spectral sequence.
The (2-primary) Adams spectral sequence [Ada58, Theorems 2.1, 2.2]computes the 2-completion of the homotopy groups of a pointed space or spectrum X . Its E -page is(3.2) E s,t = Ext s,t A ( e H ∗ ( X ; Z / , Z /
2) = ⇒ π t − s ( X ) ∧ . Here A is the 2-primary Steenrod algebra. Remark . The usual bigrading convention for Adams spectral sequences places E s,tr at coordinates ( t − s, s ).We follow this convention. The topological degree of an element at coordinates ( t − s, s ) in an Adams spectralsequence refers to t − s , and s is called its filtration .There is a general change-of-rings theorem, where if B is a graded Hopf algebra, C ⊂ B is a graded Hopfsubalgebra, and M and N are graded B -modules, then there is a natural isomorphism(3.4) Ext s,t B ( B ⊗ C M, N ) ∼ = −→ Ext s,t C ( M, N ) . When X = ko ∧ Y or ku ∧ Y , this greatly simplifies the E -page of (3.2). Inside the mod 2 Steenrodalgebra A , define the subalgebras A (1) := h Sq , Sq i and E (1) := h Q , Q i ; then, Stong [Sto63] showed e H ∗ ( ko ; Z / ∼ = A ⊗ A (1) Z / e H ∗ ( ku ; Z / ∼ = A ⊗ E (1) Z /
2. Both A (1) and E (1) areHopf subalgebras of A so (3.4) says we need only consider E s,t = Ext s,t A (1) ( e H ∗ ( X ; Z / , Z /
2) = ⇒ e ko t − s ( X ) ∧ (3.5a) E s,t = Ext s,t E (1) ( e H ∗ ( X ; Z / , Z /
2) = ⇒ f ku t − s ( X ) ∧ . (3.5b)This line of reasoning, first used by Davis [Dav74], is by now a standard trick in algebraic topology. For furtherreading, we recommend the paper of Beaudry-Campbell [BC18], who go into detail about how to define andcalculate these Ext groups and work several examples over A (1). There are fewer worked examples of (3.5b)in the literature; see Bruner-Greenlees [BG03], Nguyen [Ngu09], Francis [Fra11, §5] and Al-Boshmki [AB16]for closely related calculations.Our notation is standard in the A (1)-case, but since examples for E (1) are sparser, we record here afew notational conventions for working with E (1)-modules and this spectral sequence. When we draw E (1)-modules, we will use solid straight lines to denote Q -actions and dashed curved lines to denote Q -actions.Therefore, for example, E (1) as a module over itself looks like this.(3.6)For any E (1)-module M , H ∗ , ∗ ( E (1)) := Ext ∗ , ∗E (1) ( Z / , Z /
2) acts on Ext s,t E (1) ( M, Z / A (1)-modules; if M = e H ∗ ( X ; Z / A (1), tracking this action through the Adams spectralsequence provides information about the action of ku ∗ on f ku ∗ ( X ). Differentials are equivariant for this action,just like for the Adams spectral sequence over A (1). Since E (1) is an exterior algebra, Koszul duality providesan isomorphism of bigraded algebras(3.7) H ∗ , ∗ ( E (1)) ∼ = Z / h , v ] , where | h | = (1 ,
1) and | v | = (1 ,
3) [BC18, Example 4.5.6]. We will denote an h -action by a vertical line,and a v -action by a lighter diagonal line. Like for ko , h lifts to multiplication by 2; v lifts to the action ofthe Bott element β ∈ ku [BG03, §2.1].We will often write Ext A (1) ( M ) for Ext s,t A (1) ( M, Z / E (1); when it is clear whichsubalgebra we are working over, we will just write Ext( M ).By now there is a large body of work using the Adams spectral sequence, especially over A (1), to computethings related to invertible field theories or invertible phases. This includes [Sto86, Hil09, Fra11, FH16a,Cam17, BC18, GPW18, Guo18, FH19b, WW19a, WW19b, WWZ19, DL20a, DL20b, DL20c, GOP + These generators are given in two different bases of A ; the relations between them are Q = Sq and Q = Sq Sq +Sq Sq . NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 19
The Atiyah-Hirzebruch spectral sequence.
The (homological) Atiyah-Hirzebruch spectral sequence [AH61]for oriented bordism has signature(3.8) E p,q = e H p ( X ; Ω SO ∗ ) = ⇒ Ω SO p + q ( X ) . In general, using the Atiyah-Hirzebruch spectral sequence can feel different depending on application-specificdetails, so we point the reference-minded reader to García-Etxebarria-Montero [GEM19, §2.2.2, §3] for anintroduction and some examples which may be helpful.There are many references using the Atiyah-Hirzebruch spectral sequence to compute things related toinvertible field theories or invertible phases, such as [Edw91, Mon15, Cam17, KT17, Mon17, Hsi18, SdBKP18,SSG18, Ste18, STY18, SXG18, Xio18, ET19, FH19a, GEM19, MM19, OSS19, Shi19, TY19, BLT20, DGL20,DH20, DL20c, ETS20, GOP +
20, HH20, HKT20, Hor20, HTY20, JF20a, JF20b, KPMT20, LOT20, LT20,SFQ20, Tho20, TW20, WW20b, Yu20, DGG21].We use a few other spectral sequences in our computations, but only for one-off computations, so weaddress them when we get to them.3.3.
Adams filtration 0 phases are secretly bosonic.
In Remark 1.28, we defined a map from phasehomology with symmetry type SO to phase homology with symmetry types Spin or Spin c and interpreted itas regarding bosonic SPT phases as fermionic SPT phases in a trivial way. Physicists studying fermionicSPT phases are often interested in the cokernel of this map, which is thought of as the group of intrinsicallyfermionic SPT phases. Because bosonic crystalline phases are relatively well-understood, e.g. in the work ofHermele, Huang, Song, and their collaborators [HSHH17, SHFH17, HH18, SHQ +
19, SFQ20, SXH20] and viathe bosonic crystalline equivalence principle of Thorngren-Else [TE18], we are most interested in intrinsicallyfermionic SPT phases.The structure of the Adams spectral sequence allows us to identify the image of this bosonic-to-fermionicmap on phase homology with little extra work. For more about the Adams spectral sequence, see §3.2; fornow, we need only that phase homology groups, reinterpreted through Theorem 2.8 as groups of invertiblefield theories, are computed as homotopy groups of spectra, and that the homotopy groups of any spectrum M come with a canonical filtration called the (mod ) Adams filtration (3.9) π n M = F n ⊇ F n ⊇ F n ⊇ · · · For more information, see [BC18, §4.7]. This has two properties which are important for us.(1) The Adams spectral sequence computes the Adams filtration: after 2-completing, the associatedgraded of (3.9) is the E ∞ -page of the Adams spectral sequence, in that E s,t ∞ = gr s π t − s M .(2) If M = MTH is a Thom spectrum whose homotopy groups compute bordism groups, elements of theassociated graded in degree 0 correspond to the 2-primary part of the group of deformation classes ofinvertible TFTs which depend on something weaker than an H -structure, such as a spin IFT which isdefined by evaluating an oriented IFT on spin manifolds.This means we can identify which invertible TFTs really use the H -structure, and which do not.Now a little more detail. We do not need to say much more about (1): we depict Adams spectral sequenceson a grid with coordinates ( t − s, s ), such as in (4.24), so F n /F n is found in the E ∞ -page at coordinate ( n, n we are investigating, π n MTH is2-torsion. This assumption holds in all cases where we want to study the Adams filtration in this article,but if you want to relax it, see Remark 3.18. The assumption implies that up to extension questions on the E ∞ -page, the mod 2 Adams spectral sequence fully determines π n MTH , and that the natural map(3.10) ( π n ( MTH )) ∨ := Hom( π n ( MTH ) , C × ) −→ [ MTH , Σ n +1 I Z ]is an isomorphism.To pass from bordism groups to isomorphism class of invertible field theories, we must take character duals A A ∨ := Hom( A, C × ). This is a good thing, actually: a degree-0 element of gr • π n ( MTH ) does not usually Some extension questions can be addressed using the H ∗ , ∗ ( A (1))-action on the E ∞ -page, but there are also hiddenextensions which are harder to address. None of the calculations we make in this article manifest hidden non-split extensions;one example where they do occur is H = Spin × Z / Z / uniquely lift to an element of π n MTH : the ambiguity is F n . But in ( π n ( MTH )) ∨ , we get a subgroup: thesurjection(3.11a) π n ( MTH ) − (cid:16) π n ( MTH ) /F n ∼ = gr π n ( MTH )passes under character duality to an inclusion(3.11b) (gr π n ( MTH )) ∨ , −→ ( π n ( MTH )) ∨ . Therefore, in a mild abuse of notation, we refer to this subgroup of ( π n ( MTH )) ∨ , identified with a subgroupof the group isomorphism classes of invertible TFTs with H -structure, as the group of Adams filtration invertible TFTs with H -structure .It is a theorem [FH19b, §8.4] that this subgroup consists of theories closely related to classical Dijkgraaf-Witten theories [FQ93, §1]. Isomorphism classes of these invertible TFTs are determined by their partitionfunctions [FH16a, §5.3], so we specify these theories by their partition functions, which are bordism invariantsΩ Hn → C × .For the Adams spectral sequence, E ,n = Ext ,n A ( e H ∗ ( MTH ; Z / Z /
2) is canonically identified with(3.12) Hom A (1) ( e H ∗ ( MTH ; Z / , Σ n Z / , which is a subspace of(3.13) Hom A b ( e H n ( MTH ; Z / , Z / ∼ = ( e H n ( MTH ; Z / ∨ . The fourth quadrant of the Adams spectral sequence is empty, so E ,n ∞ is a subspace of E ,n . Take thesequence of maps(3.14a) gr π n ( MTH ) = E ,n ∞ , −→ E ,n , −→ ( e H n ( MTH ; Z / ∨ and apply character duality:(3.14b) (gr π n ( MTH )) ∨ (cid:17) − ( E ,n ) ∨ (cid:17) − e H n ( MTH ; Z / . Now compose with the Thom isomorphism to obtain(3.14c) ζ : H n ( BH ; Z / − (cid:16) (gr π n ( MTH )) ∨ . That is, a degree- n mod 2 cohomology class of BH determines an isomorphism class of Adams filtration 0invertible TFTs, and all Adams filtration 0 invertible TFTs arise in this way. The map need not be injective,e.g. by the Wu formula when H = O.Tracing this through Thom’s collapse map tells us that given a cohomology class θ ∈ H n ( BH ; Z / ζ ( θ ) is the bordism invariant which takes a closed n -manifold with H -structure( M, f : M → BH ) and returns(3.15) ζ ( θ )( M, f ) = ( − h f ∗ θ, [ M ] i . That is, use the H -structure to pull θ back to M , then evaluate it on the mod 2 fundamental class. Thisconstruction uses some aspects of the H -structure on M , but in the cases relevant to this paper, it is insensitiveto the difference between Spin and O, which is believed to pass to the physicists’ distinction between fermionicand bosonic phases. Lemma 3.16. If H = Spin × µ e G or H = Spin c × µ e G , where e G is in Data 2.1, and H := O × G , then themap H → H of tangential structures induces a surjective map H ∗ ( BH ; Z / → H ∗ ( BH ; Z / , and thereforethe partition functions (3.15) of the Adams filtration theories only depend on the underlying H -structure ofan H -manifold. These theories are not quite the same thing as classical Dijkgraaf-Witten theories, which are TFTs of oriented manifolds witha principal G -bundle, and which use R / Z -valued cohomology, rather than Z / NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 21
Proof.
First, the Spin case. We established a shearing equivalence
MTH ∼ = MTSpin ∧ X , where X is aThom spectrum of a rank-zero virtual vector bundle over BG , and this equivalence fits into a homotopycommutative diagram(3.17a) MTH ’ (cid:47) (cid:47) (cid:15) (cid:15) MTSpin ∧ X (cid:15) (cid:15) MTO ∧ ( BG ) + (cid:47) (cid:47) MTO ∧ X. Apply mod 2 cohomology and invoke the Thom isomorphism to obtain a commutative diagram(3.17b) H ∗ ( BH ; Z / H ∗ ( B Spin × BG ; Z / ∼ = (cid:111) (cid:111) H ∗ ( B O × BG ; Z / (cid:79) (cid:79) H ∗ ( B O × BG ; Z / id (cid:111) (cid:111) ξ (cid:79) (cid:79) The map H ∗ ( B O; Z / → H ∗ ( B Spin; Z /
2) is surjective, so the Künneth formula implies ξ is too, so theleft-hand arrow H ∗ ( BH ; Z / → H ∗ ( BH ; Z /
2) is as well.For H = Spin c , the proof is the same – B Spin c has an additional characteristic class c ∈ H ( B Spin c ; Z ),but its mod 2 reduction is w , so ξ is still surjective. (cid:3) Remark . In all cases that one might reasonably encounter, the bordism group π n X is finitely generated,so we can ask what happens if it contains p -torsion for an odd prime p or free summands. For a p -torsionsummand, the story is very similar: one instead uses the mod p Adams filtration on π n M ∧ p , which is detectedby the Z /p -Adams spectral sequence. This has almost the same signature as the Z / Z / Z /p and the Steenrod algebra is over Z /p instead of Z /
2. Because the mod p Thom isomorphism requires an orientation, the story is a little more nuanced fortangential structures which do not induce an orientation.For free summands in π n M , there is no analogous story. The invertible field theories in question are nottopological, and at present their classification is still a conjecture [Fre19, Lecture 9]. Assuming this conjecture,though, the Adams filtration does not tell the whole story. For example, consider 3d invertible spin fieldtheories, (conjecturally) classified by(3.19) [ MTSpin , Σ I Z ] ∼ = −→ Hom(Ω
Spin4 , Z ) ∼ = Z , generated by the map ϕ sending a spin 4-manifold to its signature divided by 16 [Roh52]. As the signaturedoes not depend on the spin structure, 16 ϕ generates Hom(Ω SO4 , Z ), and therefore the image of the forgetfulmap [ MTSO , Σ I Z ] → [ MTSpin , Σ I Z ] is identified with the subgroup 16 Z . That is, assuming the conjectureon the classification of not-necessarily-topological invertible field theories, a 3d spin invertible field theoryonly depends on the underlying orientation iff it is q times a generator, where 16 | q . So for free summands inthe abelian group of isomorphism classes of invertible field theories, the Adams filtration approach does notwork, and one must use other methods.3.4. A few utility lemmas.Definition 3.20.
Let A be an abelian group, X be a connected space, and α ∈ H ( X ; Z / A α denotes the local system on X given by the Z [ π ( X )]-module with underlying abelian group A and in which g ∈ π ( X ) acts on A by ( − α ( g ) , where we interpret α as a map π ( X ) → Z / H ( X ; Z / ∼ = Hom( π ( X ) , Z / α will be the first Stiefel-Whitney class of a vector bundle, as in the following lemma. Proposition 3.21.
Let σ → B Z / denote the tautological line bundle.(1) H k ( B Z / Z w ( σ ) ) is isomorphic to Z / in odd degrees and in even degrees.(2) If n is odd, H k ( B Z /
2; ( Z /n ) w ( σ ) ) ∼ = 0 for all k .(3) If n is even, H k ( B Z /
2; ( Z /n ) w ( σ ) ) ∼ = Z / for all k . This follows from the fact that the signature defines an isomorphism σ : Ω SO4 → Z , which follows from the fact that CP ,with signature 1, generates Ω SO4 [Tho54, Remarque following Corollaire IV.18].
Proof.
Use RP ∞ := lim −→ n RP n as our model for B Z /
2. Let A be any abelian group. Given k , choose a verylarge even m ; then, the map RP m , → B Z / H k ( RP m ; A w ( σ ) ) ∼ = → H k ( B Z / A w ( σ ) ).Since m is even, RP m is unorientable, and Z w ( σ ) is isomorphic to the orientation local system for RP m , sothere is a Poincaré duality isomorphism H k ( RP m ; A w ( σ ) ) ∼ = H m − k ( RP m ; A ). (cid:3) Lemma 3.22.
Let → H → G π → Z / → be a group extension; then H ∗ ( BG ; π ∗ Z w ( σ ) ) is -torsion.Proof. The Lyndon-Hochschild-Serre spectral sequence [Lyn48, Ser50, HS53] associated to this short exactsequence takes the form(3.23) E p,q = H p ( B Z /
2; ( H q ( BH )) w ( σ ) ) = ⇒ H p + q ( BG ; π ∗ Z w ( σ ) ) . By Proposition 3.21, the E -page is 2-torsion, so the E ∞ -page is also 2-torsion, so each H n ( BG ; π ∗ Z w ( σ ) ) isalso 2-torsion. (cid:3) We will repeatedly use the following theorem to show some differentials and extensions are trivial in theAdams spectral sequence.
Theorem 3.24 (Margolis [Mar74]) . Let B be a sub-Hopf algebra of Steenrod algebra and Y be a spectrumwith e H ∗ ( Y ; Z / ∼ = A ⊗ B Z / (so that the change-of-rings trick works for computing -completed Y -homology).For any spectrum X , there is a splitting (3.25) Y ∧ X ’ F ∨ X, where F is an Eilenberg-Mac Lane spectrum for a graded Z / -vector space and e H ∗ ( X ; Z / has no freesummands as an A -module. The upshot is that in the Adams spectral sequence for computing π ∗ ( Y ∧ X ) ∧ , the piece of the E -pagecoming from free summands of e H ∗ ( X ; Z /
2) as a B -module do not emit or receive nontrivial differentials, anddo not participate in nontrivial extensions. Lemma 3.26.
Let G be a finite group and E → BG be a rank-zero virtual vector bundle.(1) If | n , e ko n ( BG E ) ⊗ Q ∼ = H ( BG ; Q w ( E ) ) ; if (cid:45) n , e ko n ( BG E ) is torsion.(2) The same is true for f ku n ( BG E ) , except divisibility by is replaced by divisibility by .Proof. Atiyah-Hirzebruch [AH61] proved that the Chern character defines an equivalence(3.27) ch : ku ∧ H Q ’ −→ _ k ≥ Σ k H Q . The Thom isomorphism theorem establishes that e H ∗ ( BG E ; Q ) ∼ = H ∗ ( BG ; Q w ( E ) ), and since G is finite, thisvanishes above degree zero by Maschke’s theorem.The proof for ko -theory is the same, except first using the complexification map c : ko → ku : (cid:3) (3.28) ch ◦ c : ko ∧ H Q ’ −→ _ k ≥ Σ k H Q . Choosing E to be the trivial bundle shows the conclusions also hold for the torsion in e ko ∗ ( BG ) and f ku ∗ ( BG ). Lemma 3.29 (Adem-Milgram) . Fix a prime p , and let H be a subgroup of a finite group G with [ G : H ] coprime to p and P be a Sylow p -subgroup of H . Assume P is abelian and that N H ( P ) /P = N G ( P ) /P ; thenthe restriction map ρ H,G : H ∗ ( BG ; Z /p ) → H ∗ ( BH ; Z /p ) is an isomorphism.Proof. This is a slight strengthening of theorems of Swan [Swa60] and Adem-Milgram [AM04, TheoremsII.6.6 and II.6.8], who prove that if K is a finite group with Abelian p -Sylow subgroup P , then therestriction map H ∗ ( BK ; Z /p ) → H ∗ ( BP ; Z /p ) N K ( P ) is an isomorphism. In our setting, the data of P and N ( P ) /P are identical for G and H , so both restriction maps r P,G : H ∗ ( BG ; Z /p ) → H ∗ ( BP ; Z /p ) N and r P,H : H ∗ ( BH ; Z /p ) → H ∗ ( BP ; Z /p ) N are isomorphisms. Since r P,G = r P,H ◦ ρ G,H , we are done. (cid:3)
In the mixed unoriented case, Theorems 2.11 and 2.24 ask us to study Thom spectra for determinants ofrepresentations. We use the following lemma to simplify them.
NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 23
Lemma 3.30.
Let λ : G → O d be a faithful representation whose image contains a reflection and V λ → BG be the associated vector bundle. Then the splitting of the surjection (3.31) G λ (cid:47) (cid:47) O d π (cid:47) (cid:47) Z / lifts to a splitting of the Thom spectrum ( BG ) Det( V λ ) − as (3.32) ( BG ) Det( V λ ) − ’ −→ ( B Z / σ − ∨ M, and the inclusion e H ∗ ( M ; Z / , → e H ∗ (( BG ) Det( V λ ) − ; Z / is injective with image a complementary vectorspace to the subspace spanned by { U w ( V λ ) k | k ≥ } .Proof. Let g ∈ G be an element sent to λ by a reflection. Then g = 1, so the maps h g i , → G (cid:16) Z / B Z / σ − → ( BG ) Det( V λ ) − → ( B Z / σ − composing to (a map homotopy equivalent to) the identity, which splitsoff ( B Z / σ − . The image of the map e H ∗ (( B Z / σ − ; Z / → e H ∗ (( BG ) Det( V λ ) − ; Z /
2) is spanned by { U w ( V λ ) k | k ≥ } , and the image of e H ∗ ( M ; Z /
2) is a complementary subspace. (cid:3) Examples: rotations and reflections
Warmup: reflections.
The simplest example of the fermionic crystalline equivalence principle occurswhen the spatial symmetry is Z / µ ⊂ Spin d , andthere are two cases. The following principle is well-established in physics literature; see Shiozaki-Shapourian-Ryu [SSR17b] and Song-Huang-Fu-Hermele [SHFH17, §VII]. • If Z / µ do not mix (often written that the reflection squares to 1), then the classificationmatches the classification of pin + invertible field theories. • Conversely, if Z / µ do mix (often written that the reflection squares to ( − F ), the classificationmatches that of pin − invertible field theories.Condensed-matter theorists also study theories with time-reversal symmetry. Though this is also an antiunitarysymmetry that can mix with µ , the classification in terms of pin structures is opposite that of reflections:when time-reversal symmetry does not mix with fermion parity, we get pin − , and when it does mix, we get pin + .This is also well-established in physics, and is discussed by Kapustin-Thorngren-Turzillo-Wang [KTTW15],Freed-Hopkins [FH16a], and others.The difference between these two correspondences is a first hint that the fermionic crystalline equivalenceprinciple must be more complicated than the bosonic version; this point is raised by Thorngren-Else [TE18,§V.A] and Cheng-Wang [CW18, §II.C]. d Class D, spinless Class D, spin-1 / Z / Z / Z / Z / Z /
16 0 Z / ⊕ Z /
24 0 0 0
Table 1. Z / Z / MTPin + , MTPin − , and MTPin c . For this group action, the spinless and spin-1 / Class D, spinless.
When the reflection does not mix with the internal symmetry group, our ansatz isexactly that of Freed-Hopkins. In this setting, Z / R d as ( d −
1) + σ , where k denotes the rank- k trivial representation and σ denotes the sign representation. Let f D denote the equivariant local system ofsymmetry types for the class D spinless case. Arguing as in [FH19a, (3.6)], in space dimension d we see that(4.1) Ph Z / ( R d ; f D ) ∼ = [ MTSpin ∧ ( B Z / − σ , Σ d +2 I Z ] . Using (2.10a),
MTSpin ∧ ( B Z / − σ ’ MTPin + , identifying these phase homology groups as homotopygroups of the Anderson dual of MTPin + , as expected. Finally, to obtain the specific groups in Table 1, weuse the preexisting calculations of pin + bordism from [Gia73b, KT90a, KT90b].4.1.2. Class D, spin- / . Again Z / d − σ , and this time, reflection mixes with fermion parity.Let f D / denote the equivariant local system of symmetry types for this case. The associated bundle to the Z / − , so by Theorem 2.11,(4.2) Ph Z / ( R d ; f D / ) ∼ = [ MTSpin ∧ ( B Z / Det( σ ) − , Σ d +2 I Z ] . Because σ is a line bundle, Det( σ ) = σ . Using (2.10b), MTSpin ∧ ( B Z / σ − ’ MTPin − , so these phasehomology groups are identified with homotopy groups of the Anderson dual of MTPin − as predicted. Thesebordism groups are calculated in [ABP69, KT90b].4.1.3. Class A.
For spin c phases (those of Altland-Zirnbauer class A), the spinless and spin-1 / V λ is pin c , so Theorem 2.24 tells us to consider MTSpin c ∧ ( B Z / − σ in both cases, and by (2.10c),this spectrum is equivalent to MTPin c .Bahri-Gilkey [BG87a, BG87b] compute pin c bordism groups, giving us the phase homology groups inTable 1.4.1.4. Comparison with prior work.
Reflection-equivariant fermionic phases have been studied by many teamsof researchers with many methods. Their results agree with each other, and with us.
Class D, spinless:
These phases, especially the Z /
16 in d = 3, are studied by Song-Huang-Fu-Hermele [SHFH17,§V.A], Hsieh-Cho-Ryu [HCR16, §IV], Shiozaki-Shapourian-Ryu [SSR17b, §II.B, §II.D], Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §10.7], and Mao-Wang [MW20].
Class D, spin- / : Song-Huang-Fu-Hermele [SHFH17, §V.B], Shapourian-Shiozaki-Ryu [SSR17a, SSR17b],Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §10.7], and Bultinck-Williamson-Haegeman-Verstraete [BWHV17,§IX].
Class A:
These phases have been studied by Isobe-Fu [IF15], Hong-Fu [HF17], Shapourian-Shiozaki-Ryu [SSR17a, SSR17b], Song-Huang-Fu-Hermele [SHFH17, §4], and Shiozaki-Shapourian-Gomi-Ryu [SSGR18, §V].4.2.
Inversions.
Inversion symmetry is the Z / R d acting by ( x , . . . , x d ) ( − x , . . . , − x d ).This offers another relatively simple example of the FCEP, but with a new feature in the spin-1 / H ( B Z / Z /
2) specified by the extension 1 → Z / → e G → Z / → w ( λ ) + w ( λ ) arenot always equal. This does not change very much, as we explain in §4.2.2 below. d Class D, spinless Class D, spin-1 / Z / Z / Z / Z Z ⊕ Z / Z ⊕ Z /
43 0 Z / Z / ⊕ Z /
24 0 Z ⊕ Z / Z ⊕ Z / ⊕ Z / Table 2. Z / Z / d ; see thereferenced sections for which symmetry types appear. In low degrees, Beaudry-Campbell [BC18, §5.6] compute low-degree pin c bordism groups using the Adams spectral sequenceover A (1), using that MTPin c ’ MTSpin ∧ Σ − MU ∧ Σ − MO . One can also compute using the Adams spectral sequenceover E (1), as in §4.4.3; we found this to be a fun and useful exercise for getting comfortable with this variation of the Adamsspectral sequence. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 25
Class D, spinless case.
First, the case for which inversion symmetry and fermion parity do not mix.The Z / R d is a direct sum of d copies of the sign representation σ , so as a Z / R d is denoted dσ . This case is covered by Freed-Hopkins [FH19a, Example 3.5], and the phase homology groups are(4.3) [ MTSpin ∧ ( B Z / d − dσ , Σ d +2 I Z ] . The spectra
MTSpin ∧ ( B Z / d − dσ are periodic in d . Lemma 4.4. If d − d is divisible by , MTSpin ∧ ( B Z / d (1 − σ ) ’ MTSpin ∧ ( B Z / d (1 − σ ) .Proof. This is an instance of Theorem 1.39, using that spin structures satisfy the 2-out-of-3 property andthat, since 4 σ is spin, so is ( d − d )(1 − σ ). (cid:3) Thus we have only to determine
MTSpin ∧ ( B Z / d (1 − σ ) for small d . • When d = 0, we get MTSpin ∧ ( B Z / + . • When d = 1, (2.10a) tells us MTSpin ∧ ( B Z / − σ ’ MTPin + . • For d = 2, we have MTSpin ∧ ( B Z / − σ . • When d = −
1, (2.10b) gives
MTSpin ∧ Σ − ( B Z / σ − ’ MTPin − .The low-degree homotopy groups of these spectra that we need are computed by Giambalvo [Gia73b] andKirby-Taylor [KT90a, KT90b] (the pin + case); Anderson-Brown-Peterson [ABP69] and Kirby-Taylor [KT90b](the pin − case); Giambalvo [Gia73a] (the case d = 2); and Mahowald-Milgram [MM76] (the spin × Z / Class D, spin- / case. Now we consider the case where the inversion symmetry and µ ⊂ Pin − d mixas specified by the nontrivial extension 1 → µ → Z / → Z / →
1. This is not classified by w + w ofthe associated bundle to the spatial representation: in the language of §2, λ = ξ . Instead, this extensionis classified by w ( σ ) + w ( σ ) , and σ is not pin − , so if f D / denotes the class D spin-1 / R d , Theorem 2.11 computes Ph Z / ∗ ( R d ; f D / ) using the Thom spectrum of thevirtual bundle(4.5) − V (cid:1) ( σ + σ − dσ ) ∼ = − V (cid:1) ( d − − σ ) . Thus(4.6) Ph Z / ( R d ; f D / ) ∼ = [ MTSpin ∧ ( B Z / ( d − − σ ) , Σ d +2 I Z ] , and Lemma 4.4 says the domain is again 4-periodic, but differently from the spinless case. • When d = 0, we have MTSpin ∧ ( B Z / − σ . • When d = 1, we have MTSpin ∧ ( B Z / σ − ’ MTPin − . • When d = 2, we have MTSpin ∧ ( B Z / + . • When d = −
1, we have
MTSpin ∧ ( B Z / − σ ’ MTPin + .In the degrees we need, these bordism groups are computed in the same references we gave above in §4.2.1,and the relevant phase homology groups appear in Table 2. Remark . This fourfold periodicity in the tangential structure appears in a few other contexts in mathe-matical physics, such as recent work of Hason, Komargodski, and Thorngren [HKT20, §4.4] and Córdova,Ohmori, Shao, and Yan [COSY20] applying it to the study of anomalies of domain wall theories as well aswork of Tachikawa and Yonekura [TY19, §3] studying anomalies arising in string theory.4.2.3.
Class A.
In class A, whether with spinless or spin-1 / d leads us to study MTSpin c ∧ ( B Z / d − dσ . Forany vector bundle V → X , V ⊕ V ∼ = V ⊗ C , and complex vector bundles are spin c , so by Theorem 1.39,we can remove factors of 2 − σ from d − dσ without changing the Thom spectrum, so we want to study MTSpin c ∧ ( B Z / + when d is even and MTSpin c ∧ ( B Z / − σ ’ MTPin c when d is odd.We discussed pin c bordism in §4.1.3. Bahri-Gilkey [BG87a, BG87b] also compute Ω Spin c ∗ ( B Z / Smith homomorphism e Ω Spin c n ( B Z / → Ω Pin c n − , which sends a spin c manifold M and principal Campbell [Cam17, §7.8] shows this spectrum is equivalent to MT (Spin × Z / Z / Z / c/ bordism, is used in several places in recent mathematical physics literature, including [Cam17,Hsi18, FH19a, GEM19, TY19, DL20a, GOP +
20, HKT20, WW20a, MV21]. Z / P → M to the induced pin c structure on a smooth submanifold representative of the Poincarédual of w ( P ) ∈ H ( M ; Z / n ; thus we get the groups in Table 2 by applying theuniversal property (1.7) of I Z to either Ω Pin c ∗ or Ω Spin c ∗ ⊕ Ω Pin c ∗− , depending on dimension.4.2.4. Comparison with prior work.
Inversion-symmetric SPT phases are pretty well-studied, even in thefermionic case, and our phase homology calculations reproduce classifications of inversion-symmetric phasesin the literature.
Class D, spinless:
These phases are studied by Shiozaki-Xiong-Gomi [SXG18, §V.B] and Cheng-Wang [CW18,§III].
Class D, spin-1/2:
These phases are studied by You-Xu [YX14, §III], Shiozaki-Shapourian-Ryu [SSR17a,SSR17b], Cheng-Wang [CW18, §III], and Shiozaki-Xiong-Gomi [SXG18, §V.A].
Class A:
These phases are studied by You-Xu [YX14, §IV.A.3], Shiozaki-Shapourian-Ryu [SSR17b, §V.B],and Song-Huang-Fu-Hermele [SHFH17, §IV]. Shiozaki-Shapourian-Ryu also study the phases corre-sponding to the Z / k +2 summand in [ MTPin c , Σ k +3 I Z ] in arbitrary odd dimensions. Remark . Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §10.8] also study inversion-symmetric fermionicphases from a bordism-theoretic perspective, in both the spinless and spin-1 / Rotations.
We turn to the case of phases equivariant for the cyclic group C n acting by rotation on aplane. These phases have been studied by several groups of authors, and our results are consistent with priorwork; see §4.3.4 for more information.Let λ : C n → SO denote this representation and V λ → BC n be the associated vector bundle. One candirectly check that C n → SO lifts across Spin → SO iff n is odd.Class D, spinless Class D, spin-1 / d n §4.3.1 §4.3.2 §4.3.32 0 mod 4 Z ⊕ Z / ( n/ Z ⊕ Z / n ⊕ Z / Z ⊕ Z / n ⊕ Z / ( n/ Z ⊕ Z / ( n/ Z ⊕ Z / n Z ⊕ Z / n ⊕ Z / ( n/ , Z ⊕ Z /n Z ⊕ Z /n Z ⊕ Z /n ⊕ Z /n , Table 3. C n -equivariant phase homology groups for the cases in which C n acts by rotations.Classification of fermionic phases with a C n rotation symmetry. For the spinless classD case, these are classified by [ MTSpin ∧ ( BC n ) − V λ , Σ d +1 I Z ]; for spin-1 / MTSpin ∧ ( BC n ) + , Σ d +1 I Z ]; and for class A, both spinless and spin-1 /
2, by [
MTSpin c ∧ ( BC n ) + , Σ d +1 I Z ].4.3.1. Class D, spinless case.
In this case, C n does not mix with µ ⊂ Spin, and Theorem 2.11 reducesAnsatz 1.22 to the computation of [
MTSpin ∧ ( BC n ) − V λ , Σ d +2 I Z ] if n is even, or [ MTSpin ∧ ( BC n ) + ], if n is odd. Lemma 4.9. Ω SO3 ( BC n ) ∼ = Z /n , Ω SO4 ( BC n ) ∼ = Z , and Ω SO5 ( BC n ) is torsion.Proof. Compute with the Atiyah-Hirzebruch spectral sequence for oriented bordism; it collapses for p + q ≤ E -page is torsion, implying Ω SO5 ( BC n ) is torsion. (cid:3) Corollary 4.10 (Bruner-Greenlees [BG10, Example 7.3.2, §12.2.D], García-Etxebarria and Montero [GEM19,§C.2]) . For n odd, Ω Spin3 ( BC n ) ∼ = Z /n , Ω Spin4 ( BC n ) ∼ = Z , and Ω Spin5 ( BC n ) is torsion. The presence of this summand follows from the existence of a Z / k +2 summand in Ω Pin c k +2 , which is proven by Bahri-Gilkey [BG87b]. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 27
Proof.
Because n is odd, BC n is stably trivial at 2, and MTSpin → MTSO is an equivalence away from2. (cid:3)
Theorem 4.11. If n is even, e Ω Spin3 (( BC n ) − V λ ) ∼ = Z / ( n/ , e Ω Spin4 (( BC n )) − V λ ) ∼ = Z , and e Ω Spin5 (( BC n ) − V λ ) is torsion.Proof. The computation breaks into 2-primary and odd-primary pieces. The forgetful map Ω
Spin ∗ → Ω SO ∗ is anodd-primary isomorphism, and because 2 − V λ is orientable, there is a Thom isomorphism e Ω SO ∗ (( BC n ) − V λ ) ∼ =Ω SO ∗ ( BC n ). Thus, Lemma 4.9 takes care of the odd-primary part.Write n = 2 ‘ m , where m is odd. Then the map BC ‘ → BC n is a stable 2-primary equivalence, because itinduces an isomorphism on mod 2 cohomology, so for the 2-primary piece it suffices to understand the case n = 2 ‘ . Campbell [Cam17, Theorem 1.8] studies Ω Spin d (( BC ‘ ) − V λ ), obtaining Z / ‘ − when d = 3, Z when d = 4, and torsion when d = 5, which suffices. (cid:3) Class D, spin- / case. Theorem 2.11 asks us to compute [
MTSpin ∧ ( BC n ) + , Σ d +2 I Z ], which (1.7)tells us in terms of Ω Spin ∗ ( BC n ). For n odd, we already saw this in Corollary 4.10. Proposition 4.12.
Let n ≡ . Then Ω Spin3 ( BC n ) ∼ = Z / n , Ω Spin4 ( BC n ) ∼ = Z , and Ω Spin5 ( BC n ) istorsion.Proof. Inclusion BC → BC n is a 2-local equivalence, so the fact that the 2-torsion is Z / Spin ∗ ( BC ). This was originally done by Mahowald-Milgram [MM76] but has been computed in a few other places, including Mahowald [Mah82, Lemma 7.3],Bruner-Greenlees [BG10, Example 7.3.1], Siegemeyer [Sie13, Theorem 2.1.5], and García-Etxebarria andMontero [GEM19, (C.18)]. What remains is odd-primary information, which is equivalent to the odd-primarypart of oriented bordism, which we computed in Lemma 4.9. (cid:3) Proposition 4.13.
For n ≡ , Ω Spin3 ( BC n ) ∼ = Z / ⊕ Z / n , Ω Spin4 ( BC n ) ∼ = Z , and Ω Spin5 ( BC n ) istorsion.Proof. Write n = 2 ‘ m , where m is odd. As in the proof of Theorem 4.11, the 2-primary part of theanswer is detected by BC ‘ → BC n , and the odd-primary part of the answer is detected oriented bordism.Davighi-Lohitsiri [DL20a, §A.3] compute Ω Spin k ( BC ‘ ) for k ≤
6, giving the 2-primary summand, and for theodd-primary part we use Lemma 4.9. (cid:3)
Botvinnik-Gilkey-Stolz [BGS97, Theorem 2.4], Bruner-Greenlees [BG10, Example 7.3.3], and Siege-meyer [Sie13, §2.2] do special cases of this computation, by a variety of methods.4.3.3.
Class A.
The representation of C n on R by rotations is unitary (under the standard identification R = C ), hence spin c , so in both the spinless and spin-1 / MTSpin c ∧ ( BC n ) + : in thespinless case, we have a Thom isomorphism MTSpin c ∧ ( BC n ) − V λ ’ → MTSpin c ∧ ( BC n ) + , and in the spin-1 / V λ ) is trivial, so Theorem 2.24 also gives us MTSpin c ∧ ( BC n ) + . Theorem 4.14.
The first few spin c bordism groups of BC n are Ω Spin c ( BC k ) ∼ = Z Ω Spin c ( BC k +1 ) ∼ = Z Ω Spin c ( BC k ) ∼ = Z / k Ω Spin c ( BC k +1 ) ∼ = Z / (2 k + 1)Ω Spin c ( BC k ) ∼ = Z Ω Spin c ( BC k +1 ) ∼ = Z Ω Spin c ( BC k ) ∼ = Z / k ⊕ Z /k Ω Spin c ( BC k +1 ) ∼ = ( Z / (2 k + 1)) ⊕ Ω Spin c ( BC k ) ∼ = Z Ω Spin c ( BC k +1 ) ∼ = Z , and Ω Spin c ( BC n ) is torsion for all n . There are a few other computations of e Ω Spin ∗ (( BC ‘ ) − V λ ) in low degrees by other methods. For ‘ = 1, see Gi-ambalvo [Gia73b], García-Etxebarria and Montero [GEM19, (C.21)], and Freed-Hopkins [FH19a, §5]. For ‘ >
1, see Botvinnik-Gilkey [BG97, §5] and Davighi-Lohitsiri [DL20a, §A.4]; Botvinnik-Gilkey only report the orders of the bordism groups, but theircomputations show that the groups we need are cyclic. Be aware that Campbell and Davighi-Lohitsiri consider a different vectorbundle than 2 − V λ , though their calculations apply to this case. Proof.
Write n = 2 ‘ · m , where m is odd. It suffices to compute the 2-primary piece and Ω Spin c ∗ ( BC n ) ⊗ Z [1 / C ‘ → C n is stably a 2-primary equivalence, so for the 2-primary piece it suffices to determineΩ Spin c ∗ ( BC ‘ ). Bahri-Gilkey [BG87b, Theorem 1] compute these groups; when ‘ = 0 they are Ω Spin c ∗ (pt),which begins Z , 0, Z , 0, Z , 0; and when ‘ = 0 we have the same free summands as when ‘ = 0, but additionaltorsion summands: Ω Spin c ( BC ‘ ) ∼ = Z / ‘ , and Ω Spin c ( BC ‘ ) ∼ = Z / ‘ − ⊕ Z / ‘ +1 .After smashing with H Z [1 / MTSpin c → MTSO ∧ ( B U ) + is an equivalence, so MTSO ∧ ( B U ) + detects all odd-primary torsion in spin c bordism. To compute this, we use the Atiyah-Hirzebruch spectral sequence(4.15) E p,q = H p ( B U × BC n ; Ω SO q (pt)) = ⇒ Ω SO p + q ( B U × BC n ) . The Künneth theorem implies the first few homology groups of B U × BC n are H = Z , H = Z /n , H = Z , H = ( Z /n ) ⊕ , H = Z , and H = ( Z /n ) ⊕ . When we feed this to the spectral sequence (4.15), there areno nonzero differentials to or from any element in total degree p + q <
5: because Ω SO i = 0 for i = 1 , , d : E , → E , , but the splitting Ω SO ∗ ( B U × BC n ) =Ω SO ∗ (pt) ⊕ e Ω SO ∗ ( B U × BC n ) splits off the q = 0 line splits off from the rest of the spectral sequence, killingthis d . This tells the odd-primary torsion in degrees 0 through 4, and since the 5-line of the E -page istorsion, Ω SO5 ( B U × BC n ) is also torsion. (cid:3) Comparison with prior work.
Rotation-equivariant phases in class D have been studied by severalgroups, including Shiozaki-Shapourian-Ryu [SSR17b, §IV.C], Guo-Ohmori-Putrov-Wan-Wang [GOP + n = 2, and most comprehensively byCheng-Wang [CW18, §IV, §V], who consider arbitrary n and both the spinless and spin-1 / d = 2 , d = 2. This is not a discrepancy, however: many authors restrict to considering phaseswhose low-energy effective theories are expected to be topological field theories, which in the ansatz ofFreed-Hopkins [FH16a, §§5.3–5.4] amounts to considering the torsion subgroup of the classification using I Z MTH . The non-topological theories corresponding to the free summand have been discussed in a fewreferences, including Freed [Fre19, Lecture 9] and Wan-Wang [WW20a, §7.1]; at present, their mathematicaldescription remains partly conjectural.Rotation-equivariant phases in class A are studied by Shiozaki-Shapourian-Ryu [SSR17b, §IV.D], Shiozaki-Xiong-Gomi [SXG18, §V.C.1], and Lu-Vishwanath-Khalaf [LVK19]. Shiozaki-Shapourian-Ryu and Lu-Vishwanath-Khalaf’s classifications agree with us on torsion but miss the free summand as before, andShiozaki-Xiong-Gomi’s computation completely matches ours. Again, the free summand corresponds tonon-topological invertible field theories.4.4.
Rotations and reflections.
In this section, we compute the phase homology groups corresponding tophases on R d equivariant for the D n -action of rotations and reflections in a given plane. Zhang-Wang-Yang-Qi-Gu [ZWY +
20] also study these phases for d = 2 and in class D; we compare our results to theirs in §4.4.5.Let λ be the standard real 2-dimensional representation of D n and V λ → BD n be the associatedvector bundle. Let s be a reflection in D n and r a rotation through the angle 2 π/n . Then, define x, y ∈ H ( BD n ; Z /
2) = Hom( D n , Z /
2) by x ( s ‘ r m ) := ‘ mod 2(4.16a) y ( s ‘ r m ) := m mod 2 . (4.16b)In the representation λ , s ‘ r m ∈ D n acts by an orientation-reversing endomorphism iff ‘ is odd, so w ( V λ ) = x . Proposition 4.17 ([Sna13, Theorem 4.6], [Tei92, §2.3]) . (1) If n is odd, H ∗ ( BD n ; Z / ∼ = Z / x ] .(2) If n ≡ , H ∗ ( BD n ; Z / ∼ = Z / x, y, w ] / ( xy + y ) , where | w | = 2 and w = w ( V λ ) .(3) If n ≡ , H ∗ ( BD n ; Z / ∼ = Z / x, y ] . Lemma 4.18.
For n ≡ , w ( V λ ) = xy + y . NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 29
Class D, spinless Class D, spin-1 / / d n §4.4.1 §4.4.2 §4.4.3 §4.4.42 0 mod 4 ( Z / ⊕ ( Z / ⊕ Z / Z / ⊕ Z / Z / ⊕ Z / Z / ⊕ , Z / Z / ⊕ Z / ⊕ Z / ⊕ Z / ⊕ Z /
22 mod 4 ( Z / ⊕ Z / ⊕ Z / ⊕ Z / ⊕ Z / , Z /
16 0 Z / ⊕ Z / Z / ⊕ Z / Table 4. D n -equivariant phase homology groups, where D n acts through rotations andreflections. These arise as homotopy groups of Anderson duals of MTSpin ∧ X n and MTSpin c ∧ X n , where X n is one of ( BD n ) − V λ or ( BD n ) Det( V λ ) − . See §4.4 for detailsand proofs. Proof.
Since s, r n/ ∈ D n commute, there is a map j : Z / × Z / → D n sending (1 , s and (0 , r n/ .The pullback map j ∗ : H ∗ ( BD n ; Z / → H ∗ ( B Z / × B Z / Z /
2) sends x and y to linearly independentelements of H ( B Z / × B Z / Z / D n , Z / → Hom( Z / × Z / , Z /
2) given by precomposing with j . Thus j ∗ is an isomorphism on H (–; Z / BD n and B Z / × B Z /
2, the mod 2 cohomology ring is the free symmetric algebra on H (–; Z / j ∗ is an isomorphism of cohomology rings.Thus we can compute w ( V λ ) by regarding λ as a Z / × Z / ‘ ⊂ λ be the fixed locus of s , which is a subspace, and ‘ be its orthogonal complement. Then λ = ‘ ⊕ ‘ as ( Z / × Z / s and r n/ act nontrivially on ‘ ; on ‘ , s acts trivially and r n/ acts nontrivially. Thus w ( ‘ ) = 1+ j ∗ ( y ), w ( ‘ ) = 1 + j ∗ ( x ) + j ∗ ( y ), and (cid:3) (4.19) w ( j ∗ V λ ) = w ( ‘ ) + w ( ‘ ) w ( ‘ ) + w ( ‘ ) = j ∗ ( y ( x + y )) . Lemma 4.20.
Suppose n is odd and i : Z / , → D n is the inclusion of h s i . Let V → BD n be a virtualvector bundle such that w ( V ) , as an element Hom( D n , Z / , is nonzero on s . Then, the induced map ofThom spectra b ı : ( B Z / i ∗ V → ( BD n ) V is a homotopy equivalence.Proof. By the homology Whitehead theorem, it suffices to show b ı induces an isomorphism rationally and onmod p cohomology for every prime p .For mod 2 cohomology, the Thom isomorphism rewords our question to be about the map H ∗ ( BD n ; Z / → H ∗ ( B Z / Z / H ∗ ( B Z / Z /
2) and H ∗ ( BD n ; Z /
2) are abstractlyisomorphic to Z / x ] with | x | = 1; we will show i ∗ x BD n = x B Z / , implying i ∗ is a ring isomorphism. Since x isthe only nonzero degree-one element and V and i ∗ V are both unorientable, x = w ( V ) and i ∗ x = w ( i ∗ V ) = 0.Because both V λ and i ∗ V λ are unorientable, the Thom isomorphism over Q or an odd prime p uses twistedcohomology. H ∗ ( B Z / Z w ( i ∗ V λ ) ) is 2-torsion, and by Lemma 3.22, H ∗ ( BD n ; Z w ( V λ ) ) is also 2-torsion. Theuniversal coefficient theorem then implies that with Q or Z /p coefficients, both of these cohomologies vanish,and b ı vacuously induces an isomorphism. (cid:3) Class D, spinless case.
Since we are considering spinless fermions, the FCEP tells us to compute[
MTSpin ∧ ( BD n ) − V λ , Σ d +1 I Z ]. Proposition 4.21.
For n odd, MTSpin ∧ ( BD n ) − V λ ∼ = MTPin + . Thus [ MTSpin ∧ ( BD n ) − V λ , Σ I Z ] ∼ = Z / and [ MTSpin ∧ ( BD n ) − V λ , Σ I Z ] ∼ = Z / .Proof. Apply Lemma 4.20 with 2 − V λ to show ( BD n ) − V λ ’ ( B Z / − σ , then (2.10a) to get MTSpin ∧ ( B Z / − σ ’ MTPin + . The second claim follows from the calculation of low-degree pin + bordismgroups [Gia73b, KT90a, KT90b]. (cid:3) Now we turn to the case where n ≡ Theorem 4.22.
When n ≡ , the first few spin bordism groups of X n are e Ω Spin0 ( X n ) ∼ = Z / e Ω Spin1 ( X n ) ∼ = Z / e Ω Spin2 ( X n ) ∼ = Z / e Ω Spin3 ( X n ) ∼ = Z / e Ω Spin4 ( X n ) ∼ = ( Z / ⊕ e Ω Spin5 ( X n ) ∼ = Z / e Ω Spin6 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin7 ( X n ) ∼ = 0 . As usual, this together with the universal property (1.7) of I Z gives the n ≡ Proof.
We will use the Adams spectral sequence at the prime 2 to compute e Ω Spin d ( X n ) for d ≤
7. Though thisonly sees 2-primary information, it follows from Proposition 3.21 and the local coefficients Thom isomorphismthat for any odd prime p , e H ∗ ( X n ; Z ( p ) ) = 0, so the p -primary piece of e Ω Spin d ( X n ) vanishes. Thus ourcomputation suffices. Recall that w ( V λ ) = x and (from Lemma 4.18) w ( V λ ) = xy + y ; thus w (2 − V λ ) = x and w (2 − V λ ) = x + xy + y . This tells us the Steenrod squares in e H ∗ ( X n ; Z / ( U ) = U x andSq ( U ) = U ( x + xy + y ). Continuing in this vein determines the A (1)-module structure on e H ∗ ( X n ; Z /
2) inlow degrees, as shown in Figure 1. We obtain a splitting as A (1)-modules: U Uy UαU ( x y + y ) Uy Ux Uxy Ux y Uy Figure 1.
The A (1)-module structure on e H ∗ (( BD n ) − V λ ; Z /
2) in low degrees, when n ≡ α := x y + y . The submodule pictured here contains all elements ofdegree at most 7.(4.23) e H ∗ ( X n ; Z / ∼ = A (1) ⊕ Σ R ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ P. The A (1)-module R is defined to be e H ∗ (( B Z / − σ ; Z / U y . The submodule P contains no elements of degree below 8, so is irrelevant for ourlow-degree computations; we need to determine Ext( M ) for the remaining summands. For Σ k A (1), there is asingle Z / k and filtration 0, and for Σ R , see [GMM68, §2] or [BC18, Figure NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 31 E -page for this spectral sequence is(4.24) s ↑ t − s → h -equivariance and Margolis’ theorem (Theorem 3.24) forces all differentialsto vanish, and Margolis’ theorem implies there are no hidden extensions, so we are done. (cid:3) Finally, consider the case that n ≡ Theorem 4.25.
Let n ≡ . There is an r ≥ such that the first few spin bordism groups of X n are e Ω Spin0 ( X n ) ∼ = Z / e Ω Spin1 ( X n ) ∼ = Z / r e Ω Spin2 ( X n ) ∼ = Z / e Ω Spin3 ( X n ) ∼ = ( Z / ⊕ e Ω Spin4 ( X n ) ∼ = ( Z / ⊕ , and e Ω Spin5 ( X n ) is torsion.Remark . One can show that r = 2 using, e.g., the long exact sequence in homology arising from thefiber sequence Σ H Z / → τ ≤ ko τ ≤ → H Z ; we do not need this, so will not prove it. Proof of Theorem 4.25.
Using Proposition 4.17, w (2 − V λ ) = x and w (2 − V λ ) = w + x . Hence Sq ( U ) = U x and Sq ( U ) = U ( w + x ). We also need the Steenrod squares of x , y , and w . For degree reasons, Sq( x ) = x + x and Sq( y ) = y + y . Lemma 4.27 ([Mal11, §4.1]) . Sq( w ) = w + wx + w . These and the Cartan formula determine the A (1)-module structure on e H ∗ ( X n ; Z / e H ∗ ( X n ; Z / ∼ = A (1) ⊕ Σ R ⊕ Σ Z / ⊕ Σ J ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ Q ⊕ P, where P is 5-connected, and we define R , J , and Q as follows. First, R is defined to be the kernel of theaugmentation map A (1) → Z /
2; the indecomposable summand in (4.28) isomorphic to Σ R is generatedby U y and
U y . The Joker is the A (1)-module J := A (1) / (Sq ); here it is generated by U w . Finally, Q := A (1) / (Sq , Sq Sq ) and is called the upside-down question mark ; here it is generated by U w y . Foreach of these summands M in (4.28), Ext s,t A (1) ( M, Z /
2) is known in the degrees relevant to us – except for P ,which is too high-degree to affect our calculations anyways. • For Σ k A (1) there is a single Z / s = 0, t = k . • For R , J , and Q , see [BC18, Figure 29]. • For Z /
2, see [BC18, Figure 20]. The first calculations of Ext s,t A (1) ( R , Z /
2) and Ext s,t A (1) ( J, Z /
2) that we know of are due to Adams-Priddy [AP76, §3].
U UyUy UwUw Ux Uy Uw y s ↑ t − s → Figure 2.
Left: the low-degree mod 2 cohomology of ( BD n ) − V λ over A (1), n ≡ E -page of theAdams spectral sequence computing e ko ∗ (( BD n ) − V λ ). See Remark 4.26 for how to addressthe differential in topological degree 2 and Lemma 4.30 to show the differential in topologicaldegree 5 vanishes.Put these together to obtain the E -page as in Figure 2, right. Lemma 3.26 tells us the E ∞ -page is torsion,so there must be nonzero differentials in the range shown, though not necessarily the d s pictured.The first nonzero differential is a d r from the 2-line to the 1-line; by h -equivariance, it kills the entireyellow tower in the 2-line. Since a d r differential decreases t − s by 1 and increases s by r , on the E r +1 -page,the 2-line contains only the first r summands of the orange tower, and the 3-line contains only the orange Z / s = 0. There can be no further differentials to or from the 1- or 2-lines, so we obtain Z / r in degree 1 and Z / d : E , → E , . If this d = 0, there is also an extension problem in degree t − s = 4 of the form(4.29) 0 (cid:47) (cid:47) Z / (cid:47) (cid:47) e Ω Spin4 ( X n ) (cid:47) (cid:47) Z / ⊕ Z / ⊕ Z / (cid:47) (cid:47) . Lemma 4.30.
This d vanishes, and the extension (4.29) splits.Proof. We will prove this by mapping to a simpler Adams spectral sequence that has already been studied,as depicted in Figure 3.Because V λ is the pullback of the tautological bundle V → B O along Bλ : BD n → B O , we obtain amap of Thom spectra f : X n = ( BD n ) − V λ → ( B O ) − V ; the codomain is often denoted Σ MTO . Under f , our U w ∈ e H ( X n ; Z /
2) is the pullback of
U w ∈ e H (Σ MTO ).The spin bordism of Σ MTO is identified with the bordism theory of the group Pin ˜ c + := (Pin + (cid:110) Spin ) /µ .Invertible field theories for this tangential structure are believed to correspond to invertible topological phasesof Altland-Zirnbauer type AII [FH16a, (9.25), (10.2)]. Several authors study the Adams spectral sequence for Ω
Pin ˜ c + ∗ ∼ = e Ω Spin ∗ (Σ MTO ) in low degrees, in-cluding Freed-Hopkins [FH16a, Figure 5, case s = − U w ∈ e H (Σ MTO ; Z /
2) generates a Σ Z / A (1)-submodule of e H ∗ (Σ MTO ; Z / f ∗ restricts to an isomorphism from that Σ Z / Z / U w . This means the submodule of the E -page for e Ω Spin ∗ ( X n )coming from Σ Z / E -page for e Ω Spin ∗ (Σ MTO ) comingfrom the Σ Z / U w — and crucially, in that spectral sequence, E , ∼ = 0. See the commutative For further discussion, see also Metlitski [Met15] and Seiberg-Witten [SW16, §A.4].
NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 33 diagram of pink arrows in Figure 3. Thus the image of our d under f vanishes, and the map between thesespectral sequences on E , s (the targets of these d s) is an isomorphism, so our d also vanishes. Figure 3.
The map X n → Σ MTO induces a map between the Adams spectral sequencescomputing their ko -theory groups. We use this in Lemma 4.30 to show the pictured d vanishes, as the square of pink arrows in the above figure is commutative. The right-hand sideof this figure, which displays Ext( e H ∗ (Σ MTO ; Z / x, y ∈ e Ω Spin4 ( X n ) such that x = 2 y and the imageof x image in the E ∞ -page of the Adams spectral sequence is the nonzero element of E , ∞ ∼ = Z /
2. Then f mapsthis Z / Z / E ∞ -page for Σ MTO , so f ∗ ( x ) = 0. But Ω Pin ˜ c + ∼ = ( Z / ⊕ [FH16a,Theorem 9.87], so no matter where y maps to, 2 y = x
0, which is a problem. (cid:3)
We have thus determined e Ω Spin d ( X n ) for d = 3 ,
4, so we are done. (cid:3)
Class D, spin- / case. Lemma 4.31. V λ is not pin − .Proof. For n even, this follows by pulling back along BC n → BD n : we saw in §4.3 that the pullback is notspin, so V λ cannot be pin − . For n odd, pull back along the map B Z / → BD n induced by the inclusion ofa reflection; the pullback is not pin − , so neither is V λ . (cid:3) Therefore by Theorem 2.11, we consider X n := ( BD n ) Det( V λ ) − . Proposition 4.32.
For n odd, MTSpin ∧ X n ∼ = MTPin − . Thus [ MTSpin ∧ X n , Σ k +2 I Z ] vanishes for k = 2 , .Proof. Apply Lemma 4.20 with Det( V λ ) − BD n ) Det( V λ ) − ’ ( B Z / σ − , then (2.10b) to get MTSpin ∧ ( B Z / σ − ’ MTPin − . The second claim follows from the calculation of low-degree pin − bordismgroups [ABP69, KT90b]. (cid:3) Now suppose n ≡ Theorem 4.33.
The first few bordism groups of X n are e Ω Spin0 ( X n ) ∼ = Z / e Ω Spin1 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin2 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin3 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin4 ( X n ) ∼ = 0 e Ω Spin5 ( X n ) ∼ = Z / . Proof.
We show that X n ’ MTPin − ∧ ( B Z / + , so its homotopy groups are the pin − bordism groups of B Z /
2. Once we finish this, we use work of Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §7.2.1] computingΩ
Pin − k ( B Z /
2) in degrees 5 and below to conclude.
Lemma 4.34.
The inclusion i : Z / × Z / , → D n given by a reflection and a half-turn induces an equivalenceof Thom spectra ( B ( Z / × Z / i ∗ Det( V λ ) − ’ → ( BD n ) Det( V λ ) − .Proof. The map Bi : B ( Z / × Z / → BD n induces an equivalence on mod 2 cohomology, and therefore bythe Thom isomorphism theorem also induces an equivalence on the mod 2 cohomology of the Thom spectrain question.Let k be a field of characteristic not equal to 2. By Lemma 3.22 and the Thom isomorphism theorem, e H ∗ (( B ( Z / × Z / i ∗ Det( V λ ) − ; k ) and e H ∗ ( X n ; k ) both vanish, and therefore the map i induces on Thomspectra is vacuously an isomorphism on cohomology with k coefficients.Together, these allow us to use the Whitehead theorem to conclude that ( B ( Z / × Z / i ∗ Det( V λ ) − → ( BD n ) Det( V λ ) − is a homotopy equivalence. (cid:3) The stable bundle i ∗ Det( V λ ) → B ( Z / × Z /
2) splits as an exterior direct sum σ (cid:1)
0, where σ → B Z / B ( Z / × Z / i ∗ Det( V λ ) − ’ ( B Z / σ − ∧ ( B Z / + . Therefore by (2.10b), (cid:3) (4.35) MTSpin ∧ ( BD n ) Det( V λ ) − ’ MTSpin ∧ ( B Z / σ − ∧ ( B Z / + ’ MTPin − ∧ ( B Z / + . Finally, let n ≡ H ∗ ( BD n ; Z / ∼ = Z / x, y, w ] / ( xy + y ) with | x | = | y | = 1 and | w | = 2,so Sq( x ) = x + x and Sq( y ) = y + y , and from Lemma 4.27, Sq( w ) = w + wx + w . The Stiefel-Whitneyclasses of Det( V λ ) tell us that if U is the Thom class, Sq ( U ) = U x and Sq ( U ) = 0 in the cohomology of X n . Theorem 4.36.
There is an r ≥ such that the first few bordism groups of X n are e Ω Spin0 ( X n ) ∼ = Z / e Ω Spin1 ( X n ) ∼ = Z / r ⊕ Z / e Ω Spin2 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin3 ( X n ) ∼ = Z / ⊕ Z / e Ω Spin4 ( X n ) ∼ = 0 , and e Ω Spin5 ( X n ) is torsion. One can show that r = 2 using, e.g., the long exact sequence in homology arising from the fiber sequenceΣ H Z / → τ ≤ ko τ ≤ → H Z ; we do not need this, so will not prove it. Proof.
By Lemma 3.30, a choice of a reflection in D n induces a splitting(4.37) X n ’ −→ ( B Z / σ − ∨ M, such that the map e H ∗ ( M ; Z / → e H ∗ ( X n ; Z /
2) is injective with image complementary to the subspace spannedby { U x i | i ≥ } . We focus on MTSpin ∧ M , adding in the summands arising from MTSpin ∧ ( B Z / σ − ’ MTPin − at the end. The A (1)-module structure on e H ∗ ( M ; Z /
2) is determined by its image in e H ∗ ( X n ; Z / and Sq of x , y , w , and U via the Cartan formula. Using this, we draw this A (1)-module structure in Figure 4, left.As A (1)-modules,(4.38) e H ∗ ( M ; Z / ∼ = Σ R ⊕ Σ Q ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ Z / ⊕ P, where P is 5-connected, i.e. in too high degrees to affect our computations. Here Σ R is the indecomposablesummand containing U y . For the Σ k A (1) summands, we know the Ext; for Σ R , see [BC18, Figure 26],and for Σ Q , see [BC18, Figure 29]. Assembling these, we display the E -page for t − s ≤ e Ω Spin5 ( X n ) is torsion, as claimed, and that there must be a differential d r from theinfinite tower in topological degree 2 to the infinite tower in topological degree 1, though it might not bethe d pictured. Margolis’ theorem and h -equivariance rule out any other nonzero differentials to or fromelements with t − s ≤
4. Therefore in this range, E r +1 = E ∞ . The infinite tower in topological degree 2 is NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 35
Uy Uy Uy Uw Uwx Uwy Uw y s ↑ t − s → Figure 4.
Left: the A (1)-module structure on e H ∗ ( M n ; Z /
2) in low degrees. The picturedsummand contains all elements in degrees 5 and below. Right: the Ext of this module, whichis the E -page of the Adams spectral sequence converging to e ko ∗ ( M n ). See the proof ofTheorem 4.36 for more information.killed by the differential, as are all but r of the Z / M are therefore Z / r in degree 1, Z / Z / ⊕ Z / − bordism summands as computed in [ABP69, KT90b]:a Z / Z / (cid:3) Class A, spinless case.
In this case, Theorem 2.24 asks us to consider X n := MTSpin c ∧ ( BD n ) − V λ .First, assume n is odd. Recall from Proposition 4.21 that ( BD n ) − V λ ’ ( B Z / − σ and from (2.10c) that MTPin c ’ MTSpin c ∧ ( B Z / − σ . Therefore(4.39) MTSpin c ∧ ( BD n ) − V λ ’ MTPin c , and we discussed pin c bordism in §4.1.3. Theorem 4.40.
Let n ≡ ; then the low-degree spin c bordism of X n is e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ . Proof.
We use the Adams spectral sequence over E (1), which converges to f ku ∗ ( X ), together with Anderson-Brown-Peterson’s isomorphism e Ω Spin c n ( X ) ∼ = → f ku n ( X ) ⊕ f ku n − ( X ) for n ≤ A (1)-module structure on e H ∗ ( X ; Z /
2) that we calculated in (4.23) and displayed in Figure 1 determinesthe E (1)-module structure: as E (1)-modules, A (1) ∼ = E (1) ⊕ Σ E (1). Therefore(4.41) e H ∗ ( X ; Z / ∼ = E (1) ⊕ Σ R ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 7-connected; we draw a picture of this E (1)-module in Figure 5.Next Ext. For Σ k E (1), there is a unique Z / s = 0, t = k ; for Σ R , we must work alittle harder. Proposition 4.42.
Ext s,t E (1) ( R , Z / is given in Figure 6, right. U UαUy Uy Ux y Ux UβUxy Ux y Ux y Uy Figure 5.
The E (1)-module structure on e H ∗ (( BD n ) − V λ ; Z / n ≡ α := x + xy and β := x + x y . Proof.
Our proof uses as input Ext E (1) ( N ), where N is defined to be the A -module Σ − e H ∗ ( RP ; Z / Z / ; this in turn defines its A (1)- and E (1)-module structures. Davis-Mahowald [DM81, §2] calculate Ext E (1) ( N ) as a graded vector space but we also need its H ∗ , ∗ ( E (1))-modulestructure.Let h Q i ⊂ E (1) denote the subalgebra generated by Q , which is a two-dimensional vector space over Z / E (1)-modules, N ∼ = E (1) ⊗ h Q i Z /
2, so by the change-of-rings theorem (3.4), there are isomorphisms of H ∗ , ∗ ( E (1))-modules(4.43) Ext E (1) ( N ) ∼ = Ext h Q i ( Z / ∼ = Z / v ] , with v ∈ Ext , E (1) ( N , Z / h Q i is an exterior algebra.Now for R , we use the extension of E (1)-modules(4.44) 0 (cid:47) (cid:47) Σ R (cid:47) (cid:47) R (cid:47) (cid:47) N (cid:47) (cid:47) , drawn in Figure 6, left.Σ R R N s ↑ t − s → Figure 6.
Left: the extension (4.44). Right: the long exact sequence it induces of Extgroups. See the proof of Proposition 4.42 for why the long exact sequence looks like this; thekey feature is that there are no elements in odd topological degree, so all boundary mapsvanish. The dashed lines are h -extensions which are not implied by the long exact sequence,but are shown in the proof of Proposition 4.42. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 37
At first, all we know is Ext( N ). Because this lives solely in even topological degrees, and Σ R is2-connected, the long exact sequence diagram is empty in topological degree 1, so the boundary map(4.45) δ : Ext s,s +1 E (1) ( R , Z / → Ext s,s E (1) ( N , Z / t − s = 0 in Ext( R ) consists of a single Z / t − s = 2 in the long exact sequence diagram consists of two Z / N , and one in filtration 0 coming from Σ R . Since the 1-line of the diagram is empty andExt( N ) is concentrated in even degrees, the 3-line of the diagram is empty, so there are no differentials tothe 2-line. Continuing in this way produces Figure 6, right.Finally, acting by h ∈ H ∗ , ∗ ( E (1)) defines an isomorphism(4.46) Ext , E (1) ( R , Z / → Ext , E (1) ( R , Z / . This can be checked directly from the definition: begin with the unique nontrivial map R → Σ Z / h (namely the extension 0 → Σ Z / → N → Z / → (cid:3) With Ext(Σ R ) in hand, we return to our goal of computing f ku ∗ ( X ). The E -page of this Adams spectralsequence is(4.47) s ↑ t − s → n = 2 k < f ku ∗ ( X ) ∼ = ( Z / ⊕ k +1 and for n = 2 k + 1 < f ku ∗ ( X ) ∼ = Z / k +1 ; we finish with the fact that the map MTSpin c → ku ∨ Σ ku is 7-connected, so we can read off the spin c bordism groups from the ku -homologygroups. (cid:3) Theorem 4.48. If n ≡ , the first few spin c bordism groups of X n are e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ , and e Ω Spin c ( X n ) is torsion.Proof of Theorem 4.48. We again use the Adams spectral sequence over E (1); by Lemma 3.22, there is noodd-primary torsion, so this suffices. We described the A (1)-module structure on e H ∗ ( X ; Z /
2) in (4.28)and draw it in Figure 2; this determines the E (1)-module structure, with isomorphisms of E (1)-modules A (1) ∼ = E (1) ⊕ Σ E (1), R ∼ = Q ⊕ Σ E (1) and J ∼ = E (1) ⊕ Σ Z /
2. Hence as E (1)-modules,(4.49) e H ∗ ( X ; Z / ∼ = E (1) ⊕ Σ Q ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ Z / ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1)Σ Q ⊕ P, where P is 5-connected. We draw this E (1)-module in Figure 7, left.We calculated Ext( Z /
2) in (3.7), and Adams-Priddy [AP76, §3] show(4.50) Ext s,t E (1) ( Q, Z / ∼ = Ext s +1 ,t +1 E (1) ( Z / , Z / , U Ux Uy Uy UwUw Ux Uy Uw y s ↑ t − s → Figure 7.
Left: the E (1)-module structure on e H ∗ (( BD n ) − V λ ; Z / n ≡ E -page for the Adams spectral sequence computing f ku ∗ (( BD n ) − V λ ). The two pictureddifferentials are related by a v -action.with the isomorphism intertwining the H ∗ , ∗ ( E (1))-actions. We can therefore draw the E -page of the Adamsspectral sequence in Figure 7, right. We hide most v -actions to declutter the diagram. Lemma 4.51 willimply there must be differentials in this range, and Lemma 4.52 will show they are the d s pictured. Lemma 4.51. f ku ∗ ( X n ) is torsion.Proof. By Lemma 3.26, f ku ‘ ( X n ) is torsion for ‘ odd, and if ‘ is even, f ku ‘ ( X n ) ⊗ Q ∼ = H ( BD n ; Q w (2 − V λ ) ).In particular, these groups are all isomorphic, and looking at Figure 7, right, f ku ( X n ) ⊗ Q vanishes. (cid:3) Thus all of the towers must either emit or receive differentials, so as to clean up the E ∞ -page. Thedifferentials are h -equivariant, so as soon as we know the value of a differential on one element of a tower, weknow it for the whole tower; the differentials are also v -equivariant, so many differentials in higher degreesare determined by differentials in lower degrees.Because f ku ( X n ) is torsion, one of the towers in degree 2 must kill the tower in degree 1 with a d r differential. v -equivariance then implies that the tower in degree 4 kills the tower in degree 3 with a d r differential,determining f ku ( X n ) and f ku ( X n ) as in the theorem statement, but only showing that f ku ( X n ) ∼ = Z / r and f ku ( X n ) ∼ = Z / r +1 . Margolis’ theorem precludes any hidden extensions.Finally, we determine r . Lemma 4.52. f ku ( X n ) has at most elements.Proof. The map ku → H Z is an equivalence in degrees 1 and below, so f ku ( X n ) ∼ = e H ( X n ). The Thomisomorphism theorem implies e H ( X n ) ∼ = H ( BD n ; Z w ( V λ ) ), and we can compute the latter group with theLyndon-Hochschild-Serre spectral sequence [Lyn48, Ser50, HS53](4.53) E p,q = H p ( B Z / H q ( BC n ) w ( σ ) ) = ⇒ H p + q ( BD n ; Z w ( V λ ) ) , because the local system Z w ( V λ ) is the pullback of the nontrivial local system Z σ on B Z / BD n → B Z / | H ( BD n ; Z w ( V λ ) ) | is bounded above by the size ofthe 1-line of the E -page, which is (cid:3) (4.54) E , ⊕ E , ∼ = H ( B Z / Z w ( σ ) ) ⊕ H ( B Z /
2; ( Z /n ) w ( σ ) ) ∼ = Z / ⊕ Z / . Thus we obtain Z / Z / (cid:3) Class A, spin- / case. Lemma 4.55. V λ is pin c iff n is odd. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 39
Proof.
For n odd, we saw that inclusion of a reflection defines a map B Z / → BD n which is an isomorphismon mod 2 cohomology. Therefore we can compute Stiefel-Whitney classes of V λ by pulling back to B Z /
2, andwe saw that the pullback bundle is stably equivalent to a line bundle, so w = 0.For n even, recall that V λ is pin c iff β ( w ( V λ )) = 0, where β : H k (–; Z / → H k +1 (–; Z ) is the integralBockstein. The “Bock-to-Sq lemma” shows the mod 2 reduction of β ( x ) is Sq ( x ). In the notation ofProposition 4.17, for n ≡ w ( V λ ) = xy + y , and Sq ( xy + y ) = x y + xy = 0. For n ≡ w ( V λ ) = w , and by Lemma 4.27, Sq ( w ) = 0. (cid:3) Therefore for n odd, we consider X n := ( BD n ) − V λ . Lemma 4.56.
For n odd, MTSpin c ∧ X n ’ MTPin c .Proof. Lemma 4.20 produces an equivalence ( B Z / − σ ’ X n , and (2.10c) shows MTSpin c ∧ ( B Z / − σ ’ MTPin c . (cid:3) Bahri-Gilkey [BG87a, BG87b] computed pin c bordism groups, giving us the needed entries in Table 4 for n odd.For n even, Theorem 2.24 directs us to the spin c bordism of X n := ( BD n ) Det( V λ ) − . First, the case n ≡ Theorem 4.57.
The first few spin c bordism groups of X n are e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / ⊕ Z / . Because Lemma 3.26 implies e Ω Spin c ( X n ) is torsion, the phase homology groups for this symmetry type are Z / ⊕ Z / for d = 2 and Z / ⊕ Z / ⊕ Z / for d = 3 . The equivalence
MTSpin ∧ X n ’ MTPin − ∧ ( B Z / + we used in Theorem 4.33 implies an equivalence MTSpin c ∧ X n ’ MTPin c ∧ ( B Z / + , so these are also the pin c bordism groups of Z /
2. This may be ofindependent interest.
Proof.
Recall from the proof of Theorem 4.33 that ( BD n ) Det( V λ ) − ’ ( B Z / σ − ∧ ( B Z / + . Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §7.2.1] determine the A (1)-module structure on e H ∗ (( B Z / σ − ∧ B Z / Z /
2) inlow degrees. Using their work, and the isomorphisms of E (1)-modules A (1) ∼ = E (1) ⊕E (1) and R ∼ = E (1) ⊕ Σ R ,there is an isomorphism of E (1)-modules(4.58) e H ∗ (( B Z / σ − ∧ B Z / Z / ∼ = Σ E (1) ⊕ Σ R ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 4-connected. Since we began with ( B Z / σ − ∧ ( B Z / + , this does not account for everything;the disjoint basepoint gives us another summand equivalent to(4.59) MTSpin c ∧ ( B Z / σ − ’ MTPin c by (2.10c). We will add in the pin c bordism groups coming from this summand, which can be read off fromthe work of Bahri-Gilkey [BG87a, BG87b], after running the Adams spectral sequence for the other summand.Returning to (4.58), we will see momentarily that E s,t is empty when t − s = 4 and s ≥
2, which precludesdifferentials from the 5-line to the 4-line and therefore means that P does not affect the calculations we make.In Figure 8, left, we draw (4.58). We computed Ext( R ) in Proposition 4.42, so we can draw the E -page ofthe Adams spectral sequence for f ku ∗ (( B Z / σ − ∧ B Z / c bordism of ( B Z / σ − ∧ B Z / c bordism toconclude. (cid:3) Next, suppose n ≡ Uy Uy Ux y Uy s ↑ t − s → Figure 8.
Left: the E (1)-module structure on e H ∗ (( B Z / σ − ∧ B Z / Z /
2) in low degrees.The pictured submodule contains all elements in degrees 4 and below. Right: Ext of thissubmodule, which is the E -page of the Adams spectral sequence computing f ku ∗ ( M n ) for t − s ≤
4. See the proof of Theorem 4.57 for more information.
Theorem 4.60.
The first few spin c bordism groups of X n are e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / e Ω Spin c ( X n ) ∼ = ( Z / ⊕ e Ω Spin c ( X n ) ∼ = Z / ⊕ Z / ⊕ Z / . Because Lemma 3.26 implies e Ω Spin c ( X n ) is torsion, the phase homology groups for this symmetry type are ( Z / ⊕ for d = 2 and Z / ⊕ Z / ⊕ Z / for d = 3 .Proof. We closely follow the proof of Theorem 4.36. There, we established a splitting X n ’ ( B Z / σ − ∨ M n ,allowing us to focus solely on e Ω Spin ∗ ( M n ): MTSpin c ∧ ( B Z / σ − ’ MTPin c (2.10c), and we know pin c bordismgroups thanks to Bahri-Gilkey [BG87a, BG87b]. In (4.38), we determined the A (1)-module structure on e H ∗ ( M n ; Z /
2) in low degrees, and the isomorphisms of E (1)-modules R ∼ = Z / ⊕ Σ R and A (1) ∼ = E (1) ⊕ Σ E (1)mean that as E (1)-modules,(4.61) e H ∗ ( M n ; Z / ∼ = Σ Z / ⊕ Σ R ⊕ Σ Q ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ Z / ⊕ P, where P is 5-connected. Thus Ext( P ) does not affect the portion of the Adams spectral sequence for f ku ∗ ( M )in degrees t − s ≤
4, so we can ignore P . In Figure 9, left, we draw (4.61). To determine the E -page of theAdams spectral sequence, see (3.7) for Ext( Z / R ), and (4.50) for Ext( Q ). Wedraw the E -page of the Adams spectral sequence for f ku ∗ ( M n ), as in Figure 9, right — though for legibility,most v -actions are hidden. Lemma 3.26 implies there must be differentials in this range; we will showmomentarily they are the d s pictured.For π ( M n ) to be torsion, there must be a differential d r from the 2-line to the 1-line; then, π ( M n ) ∼ = Z / r ,and since Ω Pin c ∼ = 0, π ( X n ) ∼ = Z / r as well. To determine the value of r , the ≤ ku → H Z is 1-connected, so it suffices to look at e H ( X n ; Z ). This is isomorphic to H ( BD n ; Z w (Det( V λ ) − ) by theThom isomorphism theorem. Since w (Det( V λ ) −
1) = w (2 − V λ ), we can reuse Lemma 4.52 to conclude e H ( X n ; Z ) ∼ = Z /
4, so r = 2.Continuing in increasing topological degree, this d kills the entire orange tower in the 2-line, and weinfer π ( M n ) ∼ = Z /
2. The green and blue summands in the 3-line survive and split off by Margolis’ theorem. v -equivariance of differentials implies that d : E s, s → E s +2 ,s +32 is nonzero, and again maps the orangetower to the dark red tower, leaving a single Z / E , . There can be no further differentials tothe 3-line, so π ( M n ) ∼ = ( Z / ⊕ . Finally, the orange tower in the 4-line is killed by the d we most recentlydiscussed, and the two light red Z / c bordism summands back in. (cid:3) NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 41
UuUy Uw Uwx Uwy Uwx Uwy Uw y s ↑ t − s → Figure 9.
Left: the E (1)-module structure on e H ∗ ( M n ; Z /
2) in low degrees. The picturedsubmodule contains all elements in degrees 5 and below. Right: Ext of this submodule,which is the E -page of the Adams spectral sequence computing f ku ∗ ( M n ) for t − s ≤
4. Most v -actions are hidden for readability. See the proof of Theorem 4.60 for more information.4.4.5. Comparison with [ZWY + . Interacting fermionic phases equivariant for a dihedral group D n actingby rotations and reflections have also been studied by Zhang-Wang-Yang-Qi-Gu [ZWY + / n , and in Altland-Zirnbauer class D. Theyalso study systems without a spatial symmetry, using the extended supercohomology classification of Wang-Gu [WG18, WG20] to classify these phases and discuss the FCEP for dihedral groups. We find completeagreement with their results except for phases with spinless fermions when n ≡ Z / ⊕ Z / Z /
2. This appears to arise from a calculation error: as we note below inRemark 4.62, the comparison map between supercohomology and the Anderson dual of spin bordism is anisomorphism for this symmetry type.
Remark . The phases we classify are realized by the extended supercohomology classifications of Wang-Gu [WG18, WG20] and Kapustin-Thorngren [KT17]. Gaiotto-Johnson-Freyd [GJF19, §§5.4–5.6] determinethat the extended supercohomology classification à la [KT17, WG18] is the cohomology of ( BD n ) − V λ or( BD n ) Det( V λ ) − with respect to a spectrum they call fGP ×≤ , which is equivalent to the ( − I Z MTSpin . Wang-Gu’s refinement in [WG20] corresponds instead to the spectrum fGP × , equivalent tothe ( − I Z MTSpin . The connective covering maps induce comparison maps from the classifications of fermionic phases usingextended supercohomology to the classification of fermionic phases under our ansatz. For fGP × , the map issufficiently connected as to be an isomorphism between the classifications of ( d + 1)-dimensional phases forall d ≤
5. For fGP ×≤ , the map is not always an isomorphism even for d = 2: the cokernel when computingsupercohomology of X is e H ( X ; Z ), and this is nonzero e.g. for X = ( BC n ) − V λ from §4.3. But for dihedralgroups, e H (( BD n ) ξ ; Z ) vanishes whenever ξ → BD n is a rank-0 unorientable virtual vector bundle, so inthis case the comparison map is an isomorphism.5. Examples: tetrahedral, octahedral, and icosahedral symmetries
Chiral tetrahedral symmetry.
We compute phase homology groups equivariant for a chiral tetrahe-dral symmetry λ : A → SO . As far as we know, this point group has not yet been considered by physicists These classifications concern phases with an internal D n symmetry, but the fermionic crystalline equivalence principleallows us to pass back and forth. The reader may at this point wonder why our classification is a generalized homology theory, while these extendedsupercohomology classifications are generalized cohomology theories. This is a subtle point. The passage between homology andcohomology occurs because in these dimensions, we may approximate
MTSpin by KO due to Anderson-Brown-Peterson’s [ABP67]study of the connectivity of the Atiyah-Bott-Shapiro map [ABS64], then use that KO is shifted Anderson self-dual [And69,FMS07, HS14, Ric16, HLN20] to pass between I Z KO -homology and Σ KO -cohomology. See Freed-Hopkins [FH19a, §5.1] forfurther discussion. Point group Ref. Class D, spinless Class D, spin-1 / / A , T ) §5.1 0 0 0 0Pyrit. ( A × Z / T h ) §5.2 ( Z / ⊕ Z / Z / ⊕ ( Z / ⊕ Z / ⊕ ( Z / ⊕ Full tet. ( S , T d ) §5.3 Z / ⊕ ( Z / ⊕ Z / ⊕ Z / ⊕ ( Z / ⊕ Chiral oct. ( S , O ) §5.4 0 Z / Z / S × Z / O h ) §5.5 ( Z / ⊕ ( Z / ⊕ Z / ⊕ ( Z / ⊕ Z / ⊕ Z / ⊕ ( Z / ⊕ Chiral icos. ( A , I ) §5.6 0 0 0 0Full icos. ( A × Z / I h ) §5.7 ( Z / ⊕ Z / Z / ⊕ ( Z / ⊕ Z / ⊕ ( Z / ⊕ Table 5.
Phase homology groups in dimension 3 + 1 equivariant with respect to varioustetrahedral, octahedral, and icosahedral symmetries and the ways they can mix with fermionparity. See the referenced sections for how the fermionic crystalline equivalence principleassociates this data with symmetry types for invertible TFTs.in the setting of fermionic phases. We will show that our ansatz implies there are no nontrivial phases witheither spinless or spin-1 / V λ → BA denotes the vectorbundle associated to λ . Proposition 5.1. H ∗ ( BA ; Z / ∼ = Z / u, v, w ] / ( u + v + w + vw ) , where | u | = 2 and | v | = | w | = 3 . Sq( u ) = u + v + w + u , Sq( v ) = v + u + uw + v , and Sq( w ) = w + u + uv + w . Except for the Steenrod operations, this result can be found in several places, such as [Kin], so we will bebrief.
Proof sketch.
Use the Lyndon-Hochschild-Serre spectral sequence [Lyn48, Ser50, HS53] for the short exactsequence 1 → Z / × Z / → A → Z / →
1; the mod 2 cohomology of Z / H ∗ ( BA ; Z / ∼ = H ( B Z / H ∗ ( B Z / × B Z / Z / H ∗ ( B Z / × B Z / Z / Z / . We can choose this Z / Z / Z / × Z / { , α, β, α + β } by α α + β , β α , and α + β β . In a mild abuse of notation, we identify Z / × Z / H ( B Z / × B Z / Z / ∼ = Hom( Z / × Z / , Z / Z / H ∗ ( B Z / × B Z / Z / ∼ = Z / α, β ].The unique nonzero degree-2 cohomology class fixed by Z / u := α + αβ + β , and two linearlyindependent degree-3 classes fixed by Z / v := α + α β + β and w := α + αβ + β , whence therelation.For the Steenrod squares, the identification in (5.2) of H ∗ ( BA ; Z /
2) as a subalgebra of H ∗ ( B Z / × B Z / Z /
2) is the pullback map for B Z / × B Z / → BA , hence A -equivariant, so we can compute Sq( u ) in H ∗ ( B Z / × B Z / Z / α ) = α + α and Sq( β ) = β + β . (cid:3) Lemma 5.3. w ( V λ ) = 0 and w ( V λ ) = u .Proof. Since V λ is orientable, w ( V λ ) = 0, and since V λ is not spin, w ( V λ ) = 0. Since H ( BA ; Z / ∼ = Z / · u , w ( V λ ) = u . (cid:3) One way to see that this representation is not spin is to look at the binary tetrahedral group 2T , definedto be the preimage of A ⊂ SO under the double cover Spin (cid:16) SO . If V λ were spin, would be a splitextension of A by µ , but it is not split.5.1.1. Class D, spinless case. If A does not mix with the symmetry type, our ansatz reduces to thatof Freed-Hopkins, which reduces the computation of these A -equivariant phase homology groups to thecomputation of [ MTSpin ∧ ( BA ) − V λ , Σ I Z ]. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 43
Theorem 5.4.
The first few spin bordism groups of X := ( BA ) − V λ are e Ω Spin0 ( X ) ∼ = Z e Ω Spin1 ( X ) ∼ = Z / e Ω Spin2 ( X ) ∼ = 0 e Ω Spin3 ( X ) ∼ = Z / e Ω Spin4 ( X ) ∼ = Z e Ω Spin5 ( X ) ∼ = Z / ⊕ Z / e Ω Spin6 ( X ) ∼ = Z / . Thus if f D denotes the A -equivariant local system of symmetry types for this case, Ph A ( R , f D ) = 0 .Proof. At the prime 2, we use the Adams spectral sequence; if p is an odd prime, the map e Ω Spin ∗ ( X ) → e Ω SO ∗ ( X )is an isomorphism on p -torsion, and we will determine the p -torsion part of e Ω SO ∗ ( X ).First, the 2-primary piece. Letting U denote the mod 2 Thom class as usual, Sq ( U ) = 0 and Sq ( U ) = U u .This and Proposition 5.1 allow us to determine the A (1)-module structure on e H ∗ ( X ; Z /
2) in low degrees, asdepicted in Figure 10, left.
U Uw Uuv s ↑ t − s → Figure 10.
Left: the A (1)-module structure on e H ∗ (( BA ) − V λ ; Z /
2) in low degrees. Thissubmodule contains all elements of degree at most 8. Right: the E -page of the Adamsspectral sequence calculating e ko ∗ (( BA ) − V λ ), given by Ext s,t A (1) ( e H ∗ (( BA ) − V λ ; Z / , Z / A (1)-modules,(5.5) e H ∗ ( X ; Z / ∼ = Q ⊕ Σ A (1) ⊕ Σ A (1) ⊕ P, where P is 8-connected. Because we only care about degrees 6 and below, P is irrelevant for us, and for theremaining summands in (5.5), Ext s,t A (1) (– , Z /
2) has already been computed. For Σ k A (1), there’s a single Z / s = 0, t = k ; for Q , see [BC18, Figure 29]. We put this together and display the E -page for our spectralsequence in Figure 10, right. A combination of h -equivariance and Margolis’ theorem (Theorem 3.24) rulesout nontrivial differentials and hidden extensions. Therefore the 2-primary part of e Ω Spin k ( X ) has a single freesummand each in degrees 0 and 4, is 0 in degrees 1 and 2, is Z / Z / ⊕ Z / Spin ∗ → Ω SO ∗ is an equivalence after inverting 2. Moreover,because λ factors through SO , V λ → BA is orientable, so there is a Thom isomorphism e Ω SO ∗ ( X ) ∼ = → Ω SO ∗ ( BA ). Hence we just need the odd-primary part of Ω SO ∗ ( BA ), which is isomorphic to the odd-primarypart of Ω Spin ∗ ( BA ). In the degrees we care about, this is isomorphic to ko ∗ ( BA ), and Bruner-Greenlees [BG10,§7.7.E] show that the odd-primary torsion in ko ∗ ( BA ) below degree 6 consists of Z / (cid:3) Class D, spin- / case. In this case, the symmetries mix as specified by the group extension giving thebinary tetrahedral group.
Theorem 5.6.
The A -equivariant phase homology group for the class D, spin- / symmetry type in 3d istrivial.Proof. Let f D / denote the local system on R assigned to this symmetry type. Since V λ is not pin − (if it were,it would be pin − and orientable, hence spin), Theorem 2.11 says Ph A ( R ; f D / ) ∼ = [ MTSpin ∧ ( BA ) + , Σ I Z ].Bruner-Greenlees [BG10, §7.7.E] show ko ( BA ) ∼ = Z and ko ( BA ) is torsion, so this phase homology groupvanishes. (cid:3) Class A.
Let f A and f A / be the A -equivariant local systems of symmetry types for spinless, resp.spin-1 / Lemma 5.7. V λ → BA is not pin c .Proof. If β : H (–; Z / → H (–; Z ) denotes the integral Bockstein, we want to show βw ( V λ ) = 0. Lemma 5.3gives w ( V λ ) = b , and the “Bock-to-Sq lemma” says that Sq b = ab + c is equal to β ( b ) mod 2. In particular, β ( b ) = 0. (cid:3) Therefore for spin-1 / Ph A ∗ ( R ; f A / ) in terms of the spin c bordism of( BA ) Det( V λ ) − . Since V λ is orientable, this is isomorphic to the spin c bordism of BA . For spinless fermions,we use ( BA ) − V λ , as usual. Theorem 5.8.
The low-degree spin c bordism groups of X := ( BA ) − V λ and BA are e Ω Spin c ( X ) ∼ = Z Ω Spin c ( BA ) ∼ = Z e Ω Spin c ( X ) ∼ = Z / Spin c ( BA ) ∼ = Z / e Ω Spin c ( X ) ∼ = Z Ω Spin c ( BA ) ∼ = Z ⊕ Z / e Ω Spin c ( X ) ∼ = Z / ⊕ Z / Spin c ( BA ) ∼ = Z / ⊕ Z / e Ω Spin c ( X ) ∼ = Z Ω Spin c ( BA ) ∼ = Z , and in both cases, Ω Spin c is torsion. Hence both Ph A ( R ; f A ) and Ph A ( R ; f A / ) vanish.Proof. We use the equivalence
MTSpin c ’ ku ∨ Σ ku in degrees below 8, then the Adams spectral sequenceover E (1) to compute ku -homology at the prime 2.For the case of spinless fermions, use the A (1)-module structure on e H ∗ ( X ; Z /
2) from (5.5) (drawn inFigure 10, left) to compute that the E (1)-module structure is(5.9) e H ∗ ( X ; Z / ∼ = Q ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 6-connected. We draw this in Figure 11, left. We computed Ext s,t E (1) ( Q, Z /
2) in (4.50), and P istoo high-degree to be relevant to us, so the E -page of the Adams spectral sequence for f ku ∗ ( X ) is given inFigure 11, right. Margolis’ theorem (Theorem 3.24) implies this spectral sequence collapses and there are noextension problems, so we conclude.On to the spin-1 / ku ∗ ( BA ) splits as ku ∗ (pt) ⊕ f ku ∗ ( BA ), and we focus on the latter.Bruner-Greenlees [BG03, §2.6] show that 2-locally, there is an equivalence(5.10) ku ∧ BA ’ ( ku ∧ Σ BC ) ∨ _ α Σ n α H Z / α ; moreover, their calculation of ku ∗ ( BA ) [BG03, Theorem 2.6.3] impliesthe only n α < n = 2 and n = 6. This, together with Hashimoto’scomputation of f ku ∗ ( B Z /
2) [Has83, Theorem 3.1], tells us ku ∗ ( BA ) ∧ in the degrees we need. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 45
U Uw Uuw Uuv s ↑ t − s → Figure 11.
Left: the E (1)-module structure on e H ∗ (( BA ) − V λ ; Z /
2) in low degrees. The pic-ture includes all elements in degrees 6 and below. Right: Ext s,t E (1) ( e H ∗ (( BA ) − V λ ; Z / , Z / E -page of the Adams spectral sequence for f ku ∗ (( BA ) − V λ ).We still need to determine the odd-primary torsion. Since | A | = 2 ·
3, for any p >
5, the map BA → pt is a p -local equivalence, so we only have to address p = 3. In this case, Lemma 3.29 implies the inclusion j : Z / , → A as the subgroup generated by (1 2 3) induces an isomorphism H ∗ ( BA ; Z / → H ∗ ( B Z / Z / p Whitehead theorem [Ser53, Chapitre III, Théorème 3], Σ ∞ ( B Z / + → Σ ∞ ( BA ) + is a 3-primaryequivalence. Similarly, the Thom isomorphism theorem implies e H ∗ ( X ; Z / → e H ∗ (( B Z / − j ∗ V λ ; Z /
3) isan isomorphism, so at the prime 3, X ’ ( B Z / − V λ . Thus, for the purpose of computing the 3-torsion inΩ Spin c ∗ ( BA ) and e Ω Spin c ∗ ( X ), we can just work with Z / Z / j ∗ V λ is isomorphic to the direct sum of a trivial representation and the real2-dimensional representation given by rotation. Each of these is spin c , the latter because it is unitary, sothere is a Thom isomorphism MTSpin c ∧ ( B Z / − j ∗ V λ ∼ = MTSpin c ∧ ( B Z / + , so in both the spinless andspin-1 / Spin c ∗ ( B Z / (cid:3) Pyritohedral symmetry.
Pyritohedral symmetry is the action of G := A × Z / R in which A acts as the orientation-preserving symmetries of a tetrahedron and Z / λ denotethis representation and V λ → BG be the associated vector bundle. Because G splits as a direct product, it iseasier to analyze than full tetrahedral symmetry (i.e. chiral tetrahedral symmetry together with a reflection),as we will see in this and the next section.5.2.1. Spinless case.
Let X := ( BG ) − V λ . By Lemma 3.26, e Ω Spin ∗ ( X ) is 2-torsion, so we just have to workwith the Adams spectral sequence at p = 2. In the rest of this section, all cohomology is with Z / Proposition 5.11.
The first several spin bordism groups of ( BG ) − V λ are e Ω Spin0 (( BG ) − V λ ) ∼ = Z / e Ω Spin1 (( BG ) − V λ ) ∼ = 0 e Ω Spin2 (( BG ) − V λ ) ∼ = Z / e Ω Spin3 (( BG ) − V λ ) ∼ = Z / e Ω Spin4 (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin5 (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin6 (( BG ) − V λ ) ∼ = Z / ⊕ ( Z / ⊕ e Ω Spin7 (( BG ) − V λ ) ∼ = ( Z / ⊕ . Proof.
We employ a trick to reduce the amount of direct computations. We will replace (3 − V λ ) → BG with a virtual vector bundle E → BG with the same first two Stiefel-Whitney classes, but which splits as anexterior sum over BA and B Z /
2. The Thom spectrum ( BG ) E has two nice properties: the Adams E -pagefor calculating e ko ∗ (( BG ) E ) is isomorphic to that of e ko ∗ ( X ), but ( BG ) E also splits as a smash product ofThom spectra over BA and B Z /
2, simplifying the calculation of said E -page. Because we do not constructa map from e ko ∗ (( BG ) E ) to e ko ∗ (( BG ) − V λ ) or vice versa, this isomorphism does not allow us to deduce anydifferentials, but we will see that all differentials in range vanish for formal reasons, so this is no problem.The Künneth formula and Proposition 5.1 together imply(5.12) H ∗ ( BG ) ∼ = Z / x, u, v, w ] / ( u + v + w + vw ) , where | x | = 1, | u | = 2, and | v | = | w | = 3, and that Sq( x ) = x + x and the Steenrod squares of u , v , and w are as in Proposition 5.1. Lemma 5.13.
The first two Steifel-Whitney classes of V are w ( V λ ) = x and w ( V λ ) = u + x .Proof. Since this representation contains orientation-reversing symmetries, w ( V λ ) must be nonzero, so is x .For w , we saw in Lemma 5.3 that when one restricts to A ⊂ A × Z /
2, one has w ( V λ | BA ) = u ; when onerestricts to Z /
2, this is 3 copies of the sign representation, hence has w ( V λ | B Z / ) = x . (cid:3) Let E → BG be the virtual vector bundle(5.14) E := 4 − ( V λ | BA (cid:1) − σ ) , where σ → B Z / i = 1 , w i ( E ) = w i (3 − V λ ). Feeding this to the Thom isomorphism gives isomorphisms of A (1)-modules(5.15) e H ∗ (( BG ) − V λ ) ∼ = e H ∗ (( BG ) E )hence also isomorphisms of the E -pages of the corresponding Adams spectral sequences. Because E → BG is an external sum,(5.16) ( BG ) E ’ ( BA ) − V λ ∧ ( B Z / σ − . We know the A (1)-module structures on the low-degree cohomology of both summands, and the Künnethformula tells us to tensor them together (over Z /
2) to determine the A (1)-module structure on e H ∗ (( BG ) E ).In (5.5), we computed the A (1)-module structure on e H ∗ (( BA ) − V λ ) in low degrees, and split off twoΣ k A (1) summands. Margolis’ theorem (Theorem 3.24) promotes that to a splitting of spectra(5.17) ko ∧ ( BA ) − V λ ’ Σ H Z / ∨ Σ H Z / ∨ Y , such that as an A -module,(5.18) e H ∗ ( Y ) ∼ = A ⊗ A (1) ( Q ⊕ P ) , where P is 7-connected. When we smash ( B Z / σ − back in, each Σ k H Z / ∧ ( B Z / σ − contributes asummand of e H n − k (( B Z / σ − ) to e ko n (( BG ) E ), i.e. a Z / ‘ ≥ k . The upshot for A (1)-modules is(5.19) Σ k A (1) ⊗ Z / e H ∗ (( B Z / σ − ) ∼ = M ‘ ≥ k Σ ‘ H Z / . By (5.15), these summands are also present in e H ∗ (( BG ) − V λ ), and Margolis’ theorem lifts this to split offcorresponding Σ ‘ H Z / Y such that(5.20) e ko n (( BG ) E ) ∼ = π n ( Y ) ⊕ e H n − (( B Z / σ − ) ⊕ e H n − (( B Z / σ − )and as A -modules,(5.21) e H ∗ ( Y ) ∼ = A ⊗ A (1) ( Q ⊕ P ) ⊗ Z / e H ∗ (( B Z / σ − ) . The change-of-rings theorem (3.4) thus applies to the E -page of the Adams spectral sequence calculating π ∗ ( Y ), yielding(5.22) E s,t ∼ = Ext s,t A (1) (( Q ⊕ P ) ⊗ Z / e H ∗ (( B Z / σ − ) , Z / . NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 47
We will work with this spectral sequence, adding in the summands corresponding to Σ H Z / H Z / A (1)-module structure on e H ∗ (( B Z / σ − ) can be found in [BC18, Figure 4]. Lemma 5.23.
There is an isomorphism of A (1) -modules Q ⊗ Z / e H ∗ (( B Z / σ − ) ∼ = A (1) ⊕ Σ R ⊕ Σ A (1) ⊕ P ,where P is -connected.Proof. Compute directly, by hand or by computer. (cid:3)
By (5.20) and (5.21), we can work with (5.22), then add in the Z / k H Z / E -page of (5.22) is(5.24) E s,t ∼ = Ext( A (1) ⊕ Σ R ⊕ Σ A (1) ⊕ P ) . Since P is 7-connected, its Ext is concentrated in degrees irrelevant to us, and we ignore it. Ext(Σ R )is computed in the degrees we need by Beaudry-Campbell [BC18, Figures 23, 24]; using this, the E -pageof (5.22) is(5.25) s ↑ t − s → h -equivariance of differentials immediately imply there are no nontrivial differentialsor extension problems below degree 8, so we conclude. (cid:3) Class D, spin- / case. Let f D / denote the equivariant local system of symmetry types correspondingto spin-1 / f D / in terms of the spin bordism of X := ( BA × B Z / Det( V λ ) − . The isomorphismDet( V λ ) ∼ = 0 (cid:1) σ provides an isomorphism X ’ ( BA ) + ∧ ( B Z / σ − , Lemma 3.30 thus implies the spinbordism of this spectrum computes the pin − bordism of BA , which could be independently interesting. Theorem 5.26.
The first few spin bordism groups of X are e Ω Spin0 ( X ) ∼ = Z / e Ω Spin1 ( X ) ∼ = Z / e Ω Spin2 ( X ) ∼ = Z / ⊕ Z / e Ω Spin3 ( X ) ∼ = Z / ⊕ Z / e Ω Spin4 ( X ) ∼ = Z / . Since e Ω Spin5 ( X ) is torsion by Lemma 3.26, Ph A × Z / ( R ; f D / ) ∼ = Z / .Proof. Use Lemma 3.30 to split X ’ ( B Z / σ − ∧ M , where the map e H ∗ ( M ; Z / → e H ∗ ( X ; Z /
2) is injectivewith image a complimentary subspace to Z / · { U x k | k ≥ } .As usual, w (Det( V λ ) −
1) = w ( V λ ) = x and w (Det( V λ ) −
1) = 0. We also need to know the A -action on H ∗ ( BG ; Z / A -action on H ∗ ( BA ; Z / A -action on H ∗ ( B Z / Z / A (1)-module structure on e H ∗ ( M ; Z / A (1)-modules(5.27) e H ∗ ( M ; Z / ∼ = Σ R ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ P, where P is 4-connected. We will see in Figure 12, right, that for t − s ≤ E s,t is concentrated in Adamsfiltration 0; this and the 4-connectedness of P imply its contribution to the E -page cannot affect the spectral sequence in degrees t − s ≤
4, which is all we need. We draw these summands, except for P , in Figure 12,left. Uu UαU ( u x + v x + w x ) Uux Uv Uux s ↑ t − s → Figure 12.
Left: the A (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. This pictureincludes all summands in degrees 4 and below. Here α := u x + v + w . Right: the E -pageof the corresponding Adams spectral sequence.Freed-Hopkins [FH16a, Figure 5, case s = 3] and Beaudry-Campbell [BC18, Figures 32, 33] calculateExt( R ) in the range we need, and we can draw the E -page of the Adams spectral sequence in Figure 12,right. This collapses, so we add in the pin − bordism summands we need from [ABP69, KT90b] to obtain thegroups in the theorem. (cid:3) Class A, spinless case.
Let f A denote the equivariant local system of symmetry types corresponding tospinless fermions in class A and X := ( BA × B Z / − V λ ; then we saw that Ph A × Z / ( R ; f A ) is determinedby e Ω Spin c ∗ ( X ). Theorem 5.28.
The first few spin c bordism groups of X are e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = 0 e Ω Spin c ( X ) ∼ = ( Z / ⊕ e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = Z / ⊕ ( Z / ⊕ e Ω Spin c ( X ) ∼ = ( Z / ⊕ e Ω Spin c ( X ) ∼ = Z / ⊕ ( Z / ⊕ e Ω Spin c ( X ) ∼ = ( Z / ⊕ , so Ph A × Z / ( R ; f A ) ∼ = Z / ⊕ ( Z / ⊕ .Proof. The twisted Thom isomorphism and Lemma 3.22 imply e H ∗ ( X ; Z ) is 2-torsion. Therefore for any oddprime p , the mod p Whitehead theorem [Ser53, Chapitre III, Théorème 3] implies e Ω Spin c ∗ ( X ) also has no p -torsion. This leaves only p = 2, for which we use the Adams spectral sequence over E (1).We determined the A (1)-module structure on e H ∗ (( BG ) − V λ ) as given in (5.24), together with an Σ ‘ A (1) for ‘ = 3 ,
4, and two Σ ‘ A (1) summands for ‘ ≥
5. This determines the E (1)-module structure: as E (1)-modules, A (1) ∼ = E (1) ⊕ Σ E (1), and R ∼ = H , so(5.29) e H ∗ ( X ; Z / ∼ = Σ H ⊕ V ⊗ Z / E (1) ⊕ P, NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 49 where V is a graded Z / P is 7-connected. Therefore the E -page is(5.30) s ↑ t − s → (cid:3) Class A, spin- / case. To compute the ( A × Z / f A / specified by the spin-1 / c bordism of X := ( BA × B Z / Det( V λ ) − ’ ( BA × B Z / (cid:1) σ − ; we know V λ is not pin c because we saw inLemma 5.7 that the pullback of V λ along BA → BA × B Z / c . Theorem 5.31.
The first few spin c bordism groups of X are e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = 0 e Ω Spin c ( X ) ∼ = Z / ⊕ Z / e Ω Spin c ( X ) ∼ = ( Z / ⊕ e Ω Spin c ( X ) ∼ = Z / ⊕ ( Z / ⊕ By Lemma 3.26, e Ω Spin c ( X ) is torsion, so Ph A × Z / ( R ; f A / ) ∼ = Z / ⊕ ( Z / ⊕ .Proof. We reuse our work from §5.2.2. We saw that X ’ ( B Z / σ − ∨ M , and we gave the low-degreecohomology of M as an A (1)-module in (5.27) (and drew it in Figure 12, left). This determines the E (1)-module structure on it, so we can calculate spin c bordism of M using the Adams spectral sequence. For theother summand, we have MTSpin c ∧ ( B Z / σ − ’ MTPin c , so we direct-sum in the pin c bordism groupscomputed by Bahri-Gilkey [BG87a, BG87b].There are isomorphisms of E (1)-modules A (1) ∼ = E (1) ⊕ Σ E (1) and R ∼ = Σ E (1) ⊕ Σ R . Therefore as an E (1)-module,(5.32) e H ∗ ( M ; Z / ∼ = Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ R ⊕ Σ E (1) ⊕ P, where P is 4-connected. As usual for these cases, we will see that Ext( e H ∗ ( M ; Z / , Z /
2) has no nonzeroelements with t − s = 4 and s >
1, so P does not affect our calculations. See Figure 13, left, for a picture ofthe E (1)-module structure on e H ∗ ( M ; Z / R ) in Proposition 4.42 to obtain the E -page ofthe Adams spectral sequence as in Figure 13, right. This collapses, so we add in the pin c bordism summandsand conclude. (cid:3) I could get used to Adams spectral sequences like this one. But alas, they are not all this easy, as we willsee in the next section.5.3.
Full tetrahedral symmetry.
The full group of symmetries of the tetrahedron, including reflections,is the symmetric group S , acting via the representation λ : S → O , which is isomorphic to the quotient ofthe four-dimensional real permutation representation by the fixed line R · (1 , , , Proposition 5.33 ([Ngu09, §2.3]) . H ∗ ( BS ; Z / ∼ = Z / a, b, c ] / ( ac ) , with | a | = 1 , | b | = 2 , and | c | = 3 . TheSteenrod squares of the generators are Sq( a ) = a + a , Sq( b ) = b + ab + c + b , and Sq( c ) = c + bc + c . Let V λ → BS denote the associated vector bundle to λ . Uu Uux Uv Uα Uux s ↑ t − s → Figure 13.
Left: the E (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. This pictureincludes all summands in degrees 4 and below. Here α := u + vx + wx . Right: the E -pageof the corresponding Adams spectral sequence. Proposition 5.34. w ( V λ ) = a , w ( V λ ) = b , and w ( V λ ) = c .Proof. Since λ does not factor through SO ⊂ O , V λ is unorientable. Thus w ( V λ ) = 0, and a is the onlynonzero element of H ( BS ; Z / w ( V λ ) = a . For w , we calculated in Lemma 5.3 that w ( V λ | A ) = 0, so w cannot vanish in BS . Our options are a , b , and a + b . Let Z / ⊂ S be generated by a transposition; thenas a Z / λ ∼ = R ⊕ σ , so w ( V λ | Z / ) = 0. The map H ∗ ( BS ; Z / → H ∗ ( B Z / Z / ∼ = Z / x ]sends b, c a x , so the constraint w ( V λ | Z / ) = 0 rules out w ( V λ ) = a and w ( V λ ) = a + b ,forcing us to conclude w ( V λ ) = b . Finally, w ( V λ ) = c follows from the Wu formula. (cid:3) Class D, spinless case.
As usual in the spinless case for unorientable representations, the ansatz asksus to let X := ( BS ) − V λ and consider MTSpin ∧ X . Theorem 5.35.
The first few spin bordism groups of X are e Ω Spin0 ( X ) ∼ = Z / e Ω Spin1 ( X ) ∼ = 0 e Ω Spin2 ( X ) ∼ = Z / e Ω Spin3 ( X ) ∼ = Z / e Ω Spin4 ( X ) ∼ = Z / ⊕ ( Z / ⊕ e Ω Spin5 ( X ) ∼ = ( Z / ⊕ e Ω Spin6 ( X ) ∼ = ( Z / ⊕ e Ω Spin7 ( X ) ∼ = 0 . Proof.
We will again use the Adams spectral sequence over A (1); as a consequence of Lemma 3.22, e Ω Spin ∗ ( X )contains no odd-primary torsion, so this tells us everything.Our first task is to write down e H ∗ ( X ; Z /
2) as an A (1)-module in low degrees, using Proposition 5.34 todeduce w (3 − V λ ) = a and w (3 − V λ ) = a + b . We describe this A (1)-module structure in low degrees inFigure 14, left.Let Σ N denote the submodule generated by U b and U bc , which is a nontrivial extension of J by Σ J . Then there is an isomorphism(5.36) e H ∗ ( X ; Z / ∼ = A (1) ⊕ Σ N ⊕ Σ A (1) ⊕ Σ N ⊕ Σ A (1) ⊕ P, The indecomposable summand isomorphic to Σ N is generated by U b , and P has no elements in degrees below8, and therefore is irrelevant for our low-degree computations. As before, we know what a Σ k A (1) summand We propose calling N the butterfly ; it also appears in [WWZ20, Figure 16]. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 51
U Ub Ua Ub Ubc Ua b s ↑ t − s → Figure 14.
Left: the A (1)-module structure on e H ∗ (( BS ) − V λ ; Z /
2) in low degrees. Thissubmodule contains all elements of degree at most 7. Right: the E -page of the Adamsspectral sequence computing e ko ∗ (( BS ) − V λ ).contributes to the E -page. To compute Ext( N ), we use a well-known explicit (12-shifted) 4-periodic minimalresolution (5.37) N A (1) f (cid:111) (cid:111) Σ A (1) ⊕ Σ A (1) f (cid:111) (cid:111) Σ A (1) ⊕ Σ A (1) f (cid:111) (cid:111) Σ A (1) f (cid:111) (cid:111) Σ A (1) f (cid:111) (cid:111) Σ A (1) ⊕ Σ A (1) Σ f (cid:111) (cid:111) . . . Σ f (cid:111) (cid:111) The dimension of Ext s,t A (1) ( N , Z /
2) is the number of summands of Σ t A (1) in the s th module in the extension. This (shifted up by 2 for Σ N ) gives the orange summands in Figure 14, right.For Σ N , we use a convenient shortcut: the kernel of the map f in (5.37) is isomorphic to Σ N . Thus,the sequence (5.37) except for the first two terms forms a minimal resolution for Σ N , so for every s, t ≥ s,t A (1) ( N , Z / ∼ = Ext s +2 ,t +4 A (1) ( N , Z / H ∗ , ∗ ( A (1))-actions on both sides. This gives us the blue summands in Figure 14, right.Now we can draw the E -page for the Adams spectral sequence for e Ω Spin ∗ ( X ), and do so in Figure 14, right.Margolis’ theorem and h i -equivariance of differentials imply there is a single differential in this range thatcould be nonzero, namely the pictured d : E , → E , . Proposition 5.40. d : E , → E , vanishes; equivalently, e ko ( X ) has more than eight elements. We will prove this using the Atiyah-Hirzebruch spectral sequence in Theorem 5.50. After some practice with A (1)-modules, writing this minimal resolution down is straightforward, if a little tedious; wefound it a helpful exercise when learning this material and the interested reader might too. Though this minimal resolution iscertainly known, it is not explicitly written in many places; the resolution will not be televised. The H ∗ , ∗ ( A (1))-action on Ext s,t A (1) ( N , Z /
2) is a little obscure from this perspective; one can show that all h - and h -actions that could be nonzero for degree reasons are in fact nonzero, as stated in [BB96, §3] and [WWZ20, Figure 15]. Oneway to see this would be to use the long exact sequences in Ext associated to the two short exact sequences0 (cid:47) (cid:47) Σ Z / (cid:47) (cid:47) N (cid:47) (cid:47) Z / (cid:47) (cid:47) (cid:47) (cid:47) Σ N (cid:47) (cid:47) Q (cid:47) (cid:47) Z / (cid:47) (cid:47) , (5.38b)together with the fact that the boundary maps in the long exact sequences commute with the H ∗ , ∗ ( A (1))-action. Assuming Proposition 5.40 for now, there are no further differentials in the range we care about, but wemust address four extension questions in degrees 4, 5, and 6:0 (cid:47) (cid:47) Z / (cid:47) (cid:47) A (cid:47) (cid:47) Z / ⊕ Z / (cid:47) (cid:47) (cid:47) (cid:47) Z / (cid:47) (cid:47) e ko ( X ) (cid:47) (cid:47) A (cid:47) (cid:47) (cid:47) (cid:47) Z / (cid:47) (cid:47) e ko ( X ) (cid:47) (cid:47) Z / (cid:47) (cid:47) (cid:47) (cid:47) Z / (cid:47) (cid:47) e ko ( X ) (cid:47) (cid:47) Z / ⊕ Z / (cid:47) (cid:47) . (5.41d)(In fact, a priori, there are five extension problems, but Margolis’ theorem splits E , ∞ ∼ = Z / t − s = 6 line.)Both (5.41a) and (5.41c) split for the same reason. For k = 4 ,
5, assume the sequence does not split; then, e ko k ( X ) has an element x such that 2 x = 0 and if y is the image of 2 x in the E ∞ -page, then h y = 0. Thisfact lifts to a nonzero action by η ∈ ko carrying 2 x to some element z ∈ e ko k +1 ( X ) such that z = 2 ηx and z = 0, but 2 η = 0, causing a contradiction.Because (5.41a) splits and ( h · ) : E , ∞ → E , ∞ is an isomorphism, all possible extensions in (5.41b) give e ko ( X ) ∼ = Z / ⊕ ( Z / ⊕ .Lastly, (5.41d). Action by h defines isomorphisms E , ∞ → E , ∞ and E , ∞ → E , ∞ , and this lifts to imply( η · ) : e ko ( X ) → e ko ( X ) is injective, splitting (5.41d). (cid:3) We return to Proposition 5.40. Our proof strategy is to compute e ko ( X ) a different way. First, we passto τ ko -cohomology, following a strategy of Campbell [Cam17, §7.4] and Freed-Hopkins [FH19a, §5.1], byway of Lemma 5.42. We then run the Atiyah-Hirzebruch spectral sequence computing the τ ko -cohomologyof X . As input, we need e H ∗ ( X ; Z ), which we compute in Theorem 5.43 using a Lyndon-Hochschild-Serrespectral sequence [Lyn48, Ser50, HS53]. Lemma 5.42 (Campbell [Cam17, (7.35), (7.36)]) . There is a noncanonical equivalence I Z ( τ ko ) ’ Σ − τ ko.Thus, if τ e ko k ( Y ) is torsion, τ e ko k ( Y ) ∼ = τ e ko k − ( Y ) . This is a corollary of the shifted self-equivalence I Z KO ’ Σ KO [And69, Theorem 4.16]. By Lemma 3.26, e ko ( X ) ∼ = τ e ko ( X ) is torsion, so is isomorphic to τ e ko ( X ). We study this groupwith the Atiyah-Hirzebruch spectral sequence. As input, we compute e H ∗ ( X ; Z ), which the Thom isomorphismequates with H ∗ ( BS ; Z w ( V λ ) ). Theorem 5.43. H ( BS ; Z w ( V λ ) ) ∼ = 0 H ( BS ; Z w ( V λ ) ) ∼ = Z / H ( BS ; Z w ( V λ ) ) ∼ = 0 H ( BS ; Z w ( V λ ) ) ∼ = Z / ⊕ Z / H ( BS ; Z w ( V λ ) ) ∼ = Z / H ( BS ; Z w ( V λ ) ) ∼ = Z / ⊕ Z / . Proof.
By Lemma 3.22, H ∗ ( BS ; Z w ( V λ ) ) lacks odd-primary torsion, so so it suffices to determine H ∗ ( BS ; ( Z (2) ) w ( V λ ) ).Let R := Z (2) [ x ] / ( x − Z [ C ]-module in which the nontrivial element of C sends 1 x
7→ − x . As Z [ C ]-modules, R ∼ = Z (2) ⊕ ( Z (2) ) σ , so we will recover H ∗ ( BS ; ( Z (2) ) w ( V λ ) ) from H ∗ ( BS ; R ).The Lyndon-Hochschild-Serre spectral sequence(5.44) E ∗ , ∗ = H ∗ ( BC ; H ∗ ( BA ; R )) = ⇒ H ∗ ( BS ; R ) Anderson gives this proof in unpublished lecture notes; see Yosimura [Yos75, Theorem 4] for Anderson’s proof. There are atleast four additional proofs that I Z KO ’ Σ KO , due to Freed-Moore-Segal [FMS07, Proposition B.11], Heard-Stojanoska [HS14,Theorem 8.1], Ricka [Ric16, Corollary 5.8], and Hebestreit-Land-Nikolaus [HLN20, Example 2.8], all by different methods. We abuse notation slightly to let τ e ko denote reduced τ ko -cohomology, rather than (cid:94) τ ko . NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 53 is multiplicative; here S acts on R through sign : S → C and A acts trivially. R is a Z / x is in odd degree, and hence R -valued cohomology is Z × Z / { + , −} to denote the Z / Proposition 5.45 (Čadek [Čad99, Lemma 3.1]) . There is an isomorphism of Z × Z / -graded rings H ∗ ( BC ; R ) ∼ = Z (2) [ y ] / (2 y ) with | y | = (1 , − ) . Proposition 5.46 (Bruner-Greenlees [BG03, §2.6]) . There is a presentation of H ∗ ( BA ; Z (2) ) whose onlygenerators and relations below degree are generators α and β in degrees and , respectively, and relations α = 2 β = 0 . Corollary 5.47. As Z × Z / -graded rings, (5.48) H ∗ ( BA ; R ) ∼ = Z (2) [ α + , α − , β + , β − , . . . ] / (2 α ± , β ± , . . . ) where the generators and relations not displayed are in Z -degrees ≥ , | α ± | = (2 , ± ) , and | β ± | = (3 , ± ) . We can now display the E -page. Elements with + grading are colored red, and elements with − gradingare colored blue; differentials are even in this Z / α + ,α − α + y,α − y α + y ,α − y α + y ,α − y α + y ,α − y α + y ,α − y β + ,β − β + y,β − y β + y ,β − y β + y ,β − y β + y ,β − y β + y ,β − y y y y y y The map S → C admits a section given by { , (1 2) } ⊂ S , so the q = 0 line supports no nonzerodifferentials and does not participate in nontrivial extension problems. Looking just at elements graded − , weare done if we can show that d ( β − ) = α − y and d ( β + y ) = 0. Fortunately, Thomas [Tho74] has computed H ∗ ( BS ; Z (2) ): since H ( BS ; Z (2) ) ∼ = Z / ⊕ Z / d ( β + ) = 0, so the Leibniz rule implies d ( β + y ) = 0 too.And since H ( BS ; Z (2) ) ∼ = Z / d ( β − y ) = 0, so d ( β − ) = 0, hence must be α − y . (cid:3) Thus equipped, we tackle the Atiyah-Hirzebruch spectral sequence.
Theorem 5.50. | e ko ( X ) | ≥ (thus implying Proposition 5.40).Proof. After using Lemma 5.42, we want to compute τ e ko ( X ), which we attack with the Atiyah-Hirzebruchspectral sequence(5.51) E p,q = e H p ( X ; ( τ ko ) q ) = ⇒ τ e ko p + q ( X ) . Using Proposition 5.33 and Theorem 5.43 as input, the E -page is(5.52) 0 1 2 3 4 5 − − − − k -invariant; this includes all differentials shown in (5.52). Let r : H ∗ (–; Z ) → H ∗ (–; Z /
2) denote reduction mod 2 and β : H ∗ (–; Z / → H ∗ +1 (–; Z ) be the Bockstein. Then, Bruner-Greenlees [BG10, Corollary A.5.2] determine the k -invariants we need for ko -cohomology:(1) The green d : E p, → E p +2 , − is Sq ◦ r : e H p ( X ; Z ) → e H p +2 ( X ; Z / d : E p, − → E p +2 , − is Sq : e H p ( X ; Z / → e H p +2 ( X ; Z / d : E p, − → E p +3 , − is β ◦ Sq : e H p ( X ; Z / → e H p +3 ( X ; Z ).We computed the A (1)-module structure on e H ∗ ( X ; Z /
2) in (5.36) (and drew it in Figure 14, left), and r and β follow from this and a few facts we just calculated for e H ∗ ( X ; Z ). For k ≤
5, we proved 2 e H k ( X ; Z ) = 0,so r is injective in these degrees. Moreover, combining this with the “Bock-to-Sq lemma” r ◦ β = Sq , weconclude for k ≤ x ∈ e H k ( X ; Z / β Sq ( x ) = 0 iff Sq Sq ( x ) = 0.All together, these allow us to resolve almost all of the indicated differentials — a priori, we do not know β Sq ( x ) when x ∈ E , − ∼ = e H ( X ; Z / x not in the image of d : E , − → E , − , Sq ( x ) = 0, sothis is fine. We find the 1-line of the E -page has five Z / E , − , two in E , − , and two in E , − . There could be a nonzero d : E , − → E , − , but the remaining four summands are generated bypermanent cycles. (cid:3) Class D, spin- / case. As V λ is not pin − , Theorem 2.11 tells us to compute the spin bordism of X := ( BS ) Det( V λ ) − . Theorem 5.53.
The first few spin bordism groups of X are e Ω Spin0 ( X ) ∼ = Z / e Ω Spin1 ( X ) ∼ = Z / e Ω Spin2 ( X ) ∼ = Z / ⊕ Z / e Ω Spin3 ( X ) ∼ = Z / e Ω Spin4 ( X ) ∼ = 0 , and e Ω Spin5 ( X ) is torsion.Proof. We will again use the Adams spectral sequence over A (1); as a consequence of Lemma 3.22, e Ω Spin ∗ ( X )contains no odd-primary torsion, so the Adams spectral sequence calculation tells us everything.Recall the A (1)-module structure on H ∗ ( BS ; Z / ∼ = Z / a, b, c ] / ( ac ) from Propositions 5.33 and 5.34.Lemma 3.30 shows that inclusion of a transposition extends to a splitting(5.54) X ’ −→ ( B Z / σ − ∨ M, NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 55 and the map e H ∗ ( M ; Z / → e H ∗ ( X ; Z /
2) is injective, with image a complementary subspace to the span of { U a n | n ≥ } . As usual, we write down e H ∗ ( M ; Z /
2) as an A (1)-module in low degrees, using w (Det( V λ ) −
1) = a and w (Det( V λ ) −
1) = 0, and give the answer in Figure 15, left.
Ub Uab s ↑ t − s → Figure 15.
Left: The A (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. This submodulecontains all elements of degree at most 4. Right: the Ext of this module, which is thebeginning of the Adams spectral sequence computing e ko ∗ ( M ). More information in the proofof Theorem 5.53.Let Σ N denote the A (1)-submodule generated by U b ; this module is studied by Baker [Bak18, §5], whocalls it the “whiskered Joker.” There is an isomorphism of A (1)-modules(5.55) e H ∗ ( M ; Z / ∼ = Σ N ⊕ Σ A (1) ⊕ P, where P contains no elements of degree less than 4. Therefore if the 4-line of the E -page is empty, P doesnot enter into our calculations — and we will see momentarily that the 4-line is in fact empty. We knowwhat Σ A (1) summand contributes to the E -page of the Adams spectral sequence. For N , we leveragewhat we learned from N in §5.3.1. Specifically, the unique nonzero A (1)-module map A (1) → N has kernelisomorphic to Σ N , so a minimal resolution for Σ N induces a minimal resolution for N which has anadditional copy of A (1) in topological degree 0 and filtration 0, and in which everything else is shifted up onein filtration, giving the red summands in Figure 15, right.Thus the E -page for this Adams spectral sequence is as in Figure 15, right. In this range, the spectralsequence collapses. Combine this with the pin − bordism summands from [ABP69, KT90b] as usual to obtainthe groups in the theorem statement, and Lemma 3.26 finishes us off by telling us e Ω Spin5 ( X ) is torsion. (cid:3) Class A, spinless case.
In this case, the ansatz asks us to consider the spin c bordism of X := ( BS ) − V λ . Theorem 5.56.
The first few spin c bordism groups of X are e Ω Spin c ( X ) = Z / e Ω Spin c ( X ) = 0 e Ω Spin c ( X ) = ( Z / ⊕ e Ω Spin c ( X ) = 0 e Ω Spin c ( X ) = ( Z / ⊕ e Ω Spin c ( X ) = Z / . Therefore Ph S ( R , Th( f )) ∼ = ( Z / ⊕ .Proof. We will use the Adams spectral sequence over E (1) as usual — Lemma 3.22 and the Atiyah-Hirzebruchspectral sequence for Ω Spin c ∗ together imply e Ω Spin c ∗ ( X ) lacks odd-primary torsion.We use the A (1)-module structure on e H ∗ ( X ; Z /
2) that we determined in (5.36) and drew in Figure 14 todetermine the E (1)-module structure: as E (1)-modules, A (1) ∼ = E (1) ⊕ Σ E (1), and N ∼ = E (1) ⊕ Σ E (1) ⊕ Σ N , so as E (1)-modules,(5.57) e H ∗ ( X ; Z / ∼ = E (1) ⊕ Σ E (1) ⊕ Σ N ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 5-connected. We draw this in Figure 16, left. Recalling Ext E (1) ( N ) from (4.43), the E -page of U Ua Ub Ua Ub Ubc s ↑ t − s → Figure 16.
Left: the E (1)-module structure on e H ∗ (( BS ) − V λ ; Z /
2) in low degrees. Thepictured submodule contains all elements of degree at most 5. Right: the Ext of thismodule, which is the beginning of the E -page of the Adams spectral sequence computing f ku ∗ (( BS ) − V λ ).the Adams spectral sequence is in Figure 16, right. There can be no differentials in the range drawn fordegree reasons, and Margolis’ theorem (Theorem 3.24) implies there are no nontrivial extensions, either, sowe are done. (cid:3) Class A, spin- / case. Theorem 2.24 says that to compute the S -equivariant phase homology groupsin class A with spin-1 / f A / , we shouldinvestigate the spin c bordism of X := ( BS ) Det V λ − : we know V λ is not pin c because its pullback along BA → BS is not pin c , as we established in Lemma 5.7. Theorem 5.58.
The first few spin c bordism groups of X are e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = 0 e Ω Spin c ( X ) ∼ = Z / ⊕ Z / e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = Z / ⊕ ( Z / ⊕ . By Lemma 3.26, e Ω Spin c ( X ) is torsion, so Ph S ( R ; f A / ) ∼ = Z / ⊕ ( Z / ⊕ .Proof. We reuse our work from §5.3.2. First, X ’ ( B Z / σ − ∨ M , and we gave the low-degree cohomologyof M as an A (1)-module in (5.55), and drew it in Figure 15, left. This determines the E (1)-module structureon it, so we can calculate spin c bordism of M using the Adams spectral sequence. For the other summand,we have MTSpin c ∧ ( B Z / σ − ’ MTPin c , so we direct-sum in the pin c bordism groups computed byBahri-Gilkey [BG87a, BG87b].There are isomorphisms of E (1)-modules A (1) ∼ = E (1) ⊕ Σ E (1) and N ∼ = E (1) ⊕ Σ N . Therefore as an E (1)-module,(5.59) e H ∗ ( M ; Z / ∼ = Σ E (1) ⊕ Σ E (1) ⊕ Σ N ⊕ P, where P is 4-connected. As usual for these cases, we will see that Ext( e H ∗ ( M ; Z / , Z /
2) has no nonzeroelements with t − s = 4 and s >
1, so P does not affect our calculations. See Figure 17, left, for a picture of NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 57 the E (1)-module structure on e H ∗ ( M ; Z / N ) in (4.43), so can draw the E -page of Ub Uab Uα s ↑ t − s → Figure 17.
Left: the E (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees; the picturedsummands include all elements in degrees 4 and below. Here α := a b + b . Right: theExt of this module, which is the beginning of the E -page of the Adams spectral sequencecomputing f ku ∗ ( M ).the Adams spectral sequence in Figure 17, right. This collapses, so we add in the pin c bordism summandsand conclude. (cid:3) Chiral octahedral symmetry.
Let λ : S → O denote the representation as symmetries of anoctahedron and V λ → BS denote the associated vector bundle. Recall from Proposition 5.33 the mod 2cohomology of BS . Lemma 5.60. w ( V λ ) = 0 and w ( V λ ) = b .Proof. Since Im( λ ) ⊂ SO , w ( V λ ) = 0. We know w ( V λ ) restricts to u ∈ H ( BA ; Z /
2) by consideringtetrahedral symmetry inside octahedral symmetry and using Lemma 5.3, so w ( V λ ) could be a + b or b . Thefact that λ splits as σ ⊕ R when restricted to a Z / w ( V λ ) is b ,not a + b . (cid:3) By Lemma 5.7, the pullback of V λ to BA is not pin c , so V λ is not pin c , and hence V λ is also not pin − .5.4.1. Class D, spinless case.
Let f D denote the equivariant local system of symmetry types for the spinlessclass D case. Theorem 2.11 identifies(5.61) Ph S k ( R ; f D ) ∼ = [ MTSpin ∧ ( BS ) − V λ , Σ k +4 I Z ] , so we study the spin bordism of X := ( BS ) − V λ . Theorem 5.62.
There is an r ≥ such that the first few spin bordism groups of X are e Ω Spin0 ( X ) ∼ = Z e Ω Spin1 ( X ) ∼ = Z / e Ω Spin2 ( X ) ∼ = 0 e Ω Spin3 ( X ) ∼ = Z / ⊕ Z / r − e Ω Spin4 ( X ) ∼ = Z , and e Ω Spin5 ( X ) is torsion. Hence Ph S ( R ; f D ) = 0 . The Atiyah-Hirzebruch spectral sequence allows one to show r = 2, so e Ω Spin3 ( X ) ∼ = Z / ⊕ Z /
2. As usual,we will not need this, so do not prove it.
Proof.
From Propositions 5.33 and 5.34 we know the mod 2 cohomology of BS and the action of the Steenrodalgebra, and using Lemma 5.60 we can draw e H ∗ ( X ; Z /
2) as an A (1)-module in low degrees, which we do inFigure 18.Let N denote the A (1)-submodule of e H ∗ ( X ; Z /
2) generated by U and U a . Then,(5.63) e H ∗ ( X ; Z / ∼ = N ⊕ Σ A (1) ⊕ Σ N ⊕ Σ A (1) ⊕ P, UUa Ua Ub Ubc Uab Figure 18.
The A (1)-module structure on e H ∗ (( BS ) − V λ ; Z /
2) in low degrees. The picturedsubmodule contains all elements of degrees 6 and below.where P is 6-connected. We have not seen N before, and need to calculate its Ext. Fortunately, there is ashort exact sequence of A (1)-modules(5.64) 0 (cid:47) (cid:47) Σ J (cid:47) (cid:47) N (cid:47) (cid:47) Q (cid:47) (cid:47) , which induces a long exact sequence in Ext. In Figure 19, we display a picture both of this extension and ofthe Adams chart for computing the boundary map in the long exact sequence.Σ J N Q s ↑ t − s → Figure 19.
Left: the extension (5.64) of A (1)-modules. Right: the long exact sequence inExt induced from that extension.On to the E -page.(5.65) s ↑ t − s → NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 59
Because differentials must be h -equivariant, they all vanish in the range pictured except possibly for thosefrom the 4-line to the 3-line, one of which is indicated in the chart. By Lemma 3.26, e ko ( X ) ⊗ Q ∼ = e ko ( X ) ⊗ Q ,and from (5.65) the latter group is isomorphic to Q . Thus e Ω Spin4 ( X ) has exactly one free summand, so oneof the two towers in the 4-line lives to the E ∞ -page, and the other admits a nonzero d r differential to thetower in degree 3. Thus, on the 3-line of the E r +1 -page, there is a single green Z / s = 0,together with a red tower of r − Z / Z / ⊕ Z / r − in degree 3 as promised. (cid:3) Class D, spin- / case. Let f D / → R be the S -equivariant local system of symmetry types for thecase of spin-1 / Spin ∗ ( BS ). Theorem 5.66.
There is an r ≥ such that the first several spin bordism groups of BS are Ω Spin0 ( BS ) ∼ = Z Ω Spin1 ( BS ) ∼ = ( Z / ⊕ Ω Spin2 ( BS ) ∼ = ( Z / ⊕ Ω Spin3 ( BS ) ∼ = Z / ⊕ Z / r +1 , Ω Spin4 ( BS ) ∼ = Z ⊕ Z / Spin5 ( BS ) ∼ = 0Ω Spin6 ( BS ) ∼ = Z / . Therefore Ph S ( R ; f D / ) ∼ = Z / . One can use the Atiyah-Hirzebruch spectral sequence to show r = 2 in Theorem 5.66; we do not need thisso do not present the proof. Proof.
First, we use the Adams spectral sequence to determine the free and 2-primary parts. Since ko ∗ ( BS )splits as ko ∗ (pt) ⊕ e ko ∗ ( BS ), we focus on e ko ∗ ( BS ) and add the Bott-song summands in at the end. There isa section s of the parity map S → Z /
2, which stably splits BS . That is, there is a spectrum M , a map t : M → Σ ∞ BS , and a weak equivalence(5.67) ( s, t ) : Σ ∞ B Z / ∨ M ∼ −→ Σ ∞ BS . This also splits the A -module structure of e H ∗ ( BS ; Z /
2) as e H ∗ ( M ; Z / ⊕ e H ∗ ( B Z / Z / e H ∗ ( B Z / Z / e H ∗ ( M ; Z /
2) is isomorphic to a complimentary subspace of Z / · { a k | k ≥ } ⊂ e H ∗ ( BS ; Z / A -modules, hence A (1)-modules. We will run the Adams spectral sequence for e ko ∗ ( M ), and add the e ko ∗ ( B Z /
2) summands in at the end.The mod 2 cohomology of BS is given in Proposition 5.33, and the action of the Steenrod squares inProposition 5.34. We can therefore draw e H ∗ ( M ; Z /
2) as an A (1)-module in low degrees, which we do inFigure 20, left. We have(5.68) e H ∗ ( M ; Z / ∼ = Σ J ⊕ Σ Q ⊕ Σ A (1) ⊕ Σ A (1) ⊕ P, where P is 6-connected. Names of A (1)-modules are as in previous sections; for all these modules except for P , we have already seen Ext s,t A (1) (– , Z / P is irrelevant for degree reasons. We display the E -page inFigure 20, right.In the range pictured, h -equivariance of differentials implies the only possible nontrivial differentials arefrom the infinite tower in degree 4 to the infinite tower in degree 3; a d is pictured as an example. In fact,those towers must support a nonzero d r for some r ; by h -equivariance, d r is either zero for every elementof the tower in degree 4, or nonzero for every element. Hence, if all d r were zero for all r , then e ko ( BS )would contain a free summand, contradicting Lemma 3.26. Therefore there is some r ≥ d r differentials from the tower in degree 4 to the tower in degree 3 are nontrivial (not necessarily the d spictured). On the E ∞ -page, the tower in degree 4 vanishes, and only r + 1 summands of the degree-3 towerremain. Thus we have computed the 2-primary part of ko ∗ ( BS ) in degrees 6 and lower: b c a b b s ↑ t − s → Figure 20.
Left: the A (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. The submodulepictured here contains all elements of degree at most 6. Right: the corresponding Ext, whichis the E -page for the Adams spectral sequence converging to the 2-primary part of e ko ∗ ( M ). • From ko ∗ (pt), we have a Z summand in degrees 0 and 4 and a Z / • From e ko ∗ ( B Z / Z / Z / • From Figure 20, right, we have Z / Z / r +1 in degree 3, and a Z / Spin ∗ (–) → Ω SO ∗ (–) is an isomorphismon odd-primary torsion, so we just have to determine the odd-primary torsion in Ω SO k ( BS ) for k ≤
6. Usethe Atiyah-Hirzebruch spectral sequence(5.69) E p,q = H p ( BS ; Ω SO q ) = ⇒ Ω SO p + q ( BS ) . The coefficients Ω SO q are sums of copies of Z and Z /
2; since we care only about odd-primary torsion, wecan ignore the Z / Z in degree q = 0 and a Z in degree q = 4.The groups H ∗ ( BS ; Z ) are computed by Thomas [Tho74] to be Z in degree 0, Z / Z / ⊕ Z / ⊕ Z / p + q ≤ E -page is Z / ⊂ E , . It is not hit by nontrivial differentials, and there isno way to obtain odd-primary torsion using differentials on the Z summands on the E -page, so the onlyodd-primary torsion in Ω SO k ( BS ) with k ≤ Z / (cid:3) Class A.
As in the case of chiral tetrahedral symmetry, V λ does not admit a pin c structure, since wesaw in Lemma 5.7 that its pullback along BA → BS also does not admit a pin c structure. Let f A , resp. f A / , denote the equivariant local systems of spectra associated to the class A spinless, resp. spin-1 / Ph S ( R ; f A ) and Ph S ( R ; f A / ) in terms of the spin c bordism of ( BS ) − V λ forspinless fermions and BS for spin-1 / NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 61
Theorem 5.70.
There are integers r, r ≥ such that the low-degree spin c bordism groups of ( BS ) − V λ and BS are e Ω Spin c (( BS ) − V λ ) ∼ = Z Ω Spin c ( BS ) ∼ = Z e Ω Spin c (( BS ) − V λ ) ∼ = Z / Spin c ( BS ) ∼ = Z / e Ω Spin c (( BS ) − V λ ) ∼ = Z Ω Spin c ( BS ) ∼ = Z ⊕ Z / e Ω Spin c (( BS ) − V λ ) ∼ = Z / ⊕ Z / r Ω Spin c ( BS ) ∼ = Z / ⊕ Z / r e Ω Spin c (( BS ) − V λ ) ∼ = Z Ω Spin c ( BS ) ∼ = Z ⊕ Z / e Ω Spin c (( BS ) − V λ ) ∼ = Z / r − ⊕ Z / ⊕ ( Z / ⊕ Ω Spin c ( BS ) ∼ = Z / ⊕ Z / ⊕ Z / r +1 e Ω Spin c (( BS ) − V λ ) ∼ = Z Ω Spin c ( BS ) ∼ = Z ⊕ ( Z / ⊕ . One can use the Atiyah-Hirzebruch spectral sequence to show r = r = 2. We do not need this, so do notgo into the details. Proof.
As usual, the calculation separates into odd-primary and 2-primary parts.
Lemma 5.71.
The only odd-primary torsion in the spin c bordism of ( BS ) − V λ and BS in degrees andbelow consists of two Z / summands in degrees and .Proof. Since | S | = 2 ·
3, we only have to check 3-torsion: if ‘ ≥ BS → pt and( BS ) − V → pt are stable ‘ -primary equivalences by the Whitehead theorem [Ser53, Chapitre III, Théorème3]. The forgetful map MTSpin c → MSO ∧ ( B U ) + is an odd-primary equivalence, and since 3 − V λ isorientable, there is a Thom isomorphism(5.72) MSO ∧ ( B U ) + ∧ ( BS ) − V ’ −→ MSO ∧ ( B U ) + ∧ ( BS ) + , so in both the spinless and spin-1 / Spin c ∗ ( B U × BS ). As thehomology of B U is torsion-free, the Künneth map H ∗ ( B U ) ⊗ H ∗ ( BS ) → H ∗ ( B U × BS ) is an isomorphismof graded abelian groups. Using this together with Thomas’ [Tho74] calculation of H ∗ ( BS ), we concludethat the only odd-primary torsion in H ∗ ( B U × BS ) in degrees below 7 is Z / ⊂ H ( B U × BS ) and Z / ⊂ H ( B U × BS ).Now feed this to the Atiyah-Hirzebruch spectral sequence with signature(5.73) E p,q = H p ( B U × BS , Ω SO q (pt)) = ⇒ Ω SO p + q ( B U × BS ) . The coefficients are sums of Z and Z /
2; since we only care about 3-torsion, we can ignore the Z / E -page in total degree less than 7 is a single Z / E , and E , , coming from ourcalculation above of 3-torsion in homology. These 3-torsion summands cannot participate in any nonzerodifferentials: they do not map to each other, and cannot receive any differentials from free summands, or fromthe 7-line (which we have not calculated). Thus they persist to the E ∞ -page. It is a priori possible more3-torsion is created from free summands on the E -page, which could happen if a differential maps from afree summand to another free summand. All free summands are in even total degree, though, so this does nothappen, and the only 3-torsion in Ω SO k ( B U × BS ), for k <
7, is two Z / (cid:3) Next, we compute the 2-torsion using the Adams spectral sequence over E (1).For the spinless case, recall from (5.63) (drawn in Figure 18) the calculation of e H ∗ (( BS ) − V λ ; Z /
2) as an A (1)-module. There are isomorphisms of E (1)-modules N ∼ = Q ⊕ Σ E (1) ⊕ Σ Z / A (1) ∼ = E (1) ⊕ Σ E (1),so as E (1)-modules,(5.74) e H ∗ (( BS ) − V λ ; Z / ∼ = Q ⊕ Σ E (1) ⊕ Σ Z / ⊕ Σ E (1) ⊕ Σ Q ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 6-connected. We draw this in Figure 21, left.To draw the E -page of the Adams spectral sequence, use the computations of Ext( Q ) from (4.50) andExt E (1) ( Z /
2) from (3.7) to obtain Figure 21, right. For clarity, we do not draw most v -actions. There maybe differentials in this range, though we do not determine whether they are the d s pictured.From Figure 21, right, f ku (( BS ) − V λ ) ∼ = Z , so Lemma 3.26 implies there is a single free summand in eacheven degree and the odd-degree ku -groups are torsion. Therefore, one of the towers on the 4-line must admit U Ua Uab Ua Ub Uα Ubc Uab s ↑ t − s → Figure 21.
Left: the E (1)-module structure on e H ∗ (( BS ) − V λ ; Z /
2) in low degrees. Thepictured submodule contains all elements of degrees at most 6. Here α := a + a b . Right: thecorresponding Ext, which is the E -page of the Adams spectral sequence for f ku ∗ (( BS ) − V ).Some nonzero v -actions are hidden for clarity.a nontrivial d r differential to the tower on the 3-line, and in fact, v -equivariance of the differentials impliesthat tower on the 4-line must be the blue one coming from Σ Q . The remaining tower must survive, so onthe E ∞ -page, the 3-line has its Z / Z / r summand coming from the red tower, and the4-line has a single Z summand left. The results on f ku and f ku follow from v - and h -equivariance of d r .On to the spin-1 / ku ∗ ( BS ) ∼ = ku ∗ (pt) ⊕ f ku ∗ ( BS ). In the proof of Theorem 5.66, wesplit Σ ∞ BS ’ Σ ∞ B Z / ∨ M and determined the A (1)-module structure on e H ∗ ( M ; Z / E (1)-module structure on e H ∗ ( BS ; Z / E (1)-modules,(5.75) e H ∗ ( M ; Z / ∼ = Σ E (1) ⊕ Σ Q ⊕ Σ Z / ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 6-connected. We draw this in Figure 22, left. b c b a b a b b s ↑ t − s → Figure 22.
Left: the E (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. The picturedsubmodule contains all elements of degrees at most 6. Right: the corresponding Ext, whichis the E -page of the Adams spectral sequence computing f ku ∗ ( M ). Some v -actions arehidden to declutter the diagram.For each of these modules N (except P , which as usual is too high-degree to be relevant), we alreadycalculated Ext s,t E (1) ( N, Z / Q , see (4.50), and for Z /
2, see (3.7). Therefore the E -page for the Adams NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 63 spectral sequence is as drawn in Figure 22, right. Most of the v -actions are hidden to make the diagramclearer. We indicate locations of some possible differentials, but they are not necessarily d s.Lemma 3.26 implies f ku ∗ ( BS ) is torsion, so all towers present on the E -page must emit or receivedifferentials. Thus there is some r ≥ d r emergingfrom the orange tower on the 4-line; therefore on the E ∞ -page, the 4-line contains only the Z / E , ∞ , and the 3-line contains r Z / f ku and f ku , v -equivarianceof d r determines the E ∞ -page in the same way.It remains to add in the summands corresponding to ku ∗ (pt) and f ku ∗ ( B Z / Z summand in each even dimension, and the latter contributes Z / Z / Z / (cid:3) Full octahedral symmetry.
The full group of symmetries of the octahedron, including orientation-reversing ones, is isomorphic to G := A × Z /
2. Let λ : G → O denote the corresponding three-dimensionalreal representation of G , and V λ → BG denote the associated vector bundle. We saw in §5.4 the pullback of V λ along BS → BG is not pin c , so V λ is also not pin c , and therefore is also not pin − .The Künneth formula and Proposition 5.33 together imply(5.76) H ∗ ( BG ; Z / ∼ = Z / x, a, b, c ] / ( ac ) , where | x | = | a | = 1, | b | = 2, and | c | = 3. Lemma 5.77. w ( V λ ) = x and w ( V λ ) = b + x .Proof. For w , we know w ( V λ ) = 0 because V λ is unorientable, but because V λ | BS is orientable, w ( V λ )cannot be a or x + a , leaving w ( V λ ) = x .For w , we know the pullback of V λ to BS has w ( V | S ) = b . If i : B Z / → BG is induced by theinclusion of a reflection in G , then i ∗ λ decomposes as a direct sum of three copies of the sign representation,so i ∗ V λ ∼ = 3 σ . Therefore i ∗ w ( V λ ) = x , uniquely constraining w ( V λ ) = b + x . (cid:3) Class D, spinless case.
The FCEP says we should study the spin bordism of ( BG ) − V λ . We will argueas we did in the case of pyritohedral symmetry in §5.2, replacing 3 − V λ with a virtual vector bundle whoseAdams E -page is isomorphic to that of ( BG ) − V λ , but which is easier to calculate. This isomorphism didnot come from a map of spectra, so cannot tell us anything about differentials or hidden extensions, butjust as for pyritohedral symmetry, we will see that for entirely formal reasons, all differentials vanish andall hidden extensions split in the range we need. By Lemma 3.22, e H ∗ (( BG ) − V λ ) contains no odd-primarytorsion, so neither does e Ω Spin ∗ (( BG ) − V λ ), so using the 2-primary Adams spectral sequence suffices.For the rest of this section, all homology and cohomology is understood to be with Z / Lemma 5.78.
Let E → BG denote the virtual vector bundle induced from the virtual representation (5.79) 2 − ( V λ | S (cid:1) ( − σ )) . Then, there is an isomorphism of A (1) -modules e H ∗ (( BG ) − V λ ) ∼ = e H ∗ (( BG ) E ) , hence an isomorphism betweenthe E -pages of the Adams spectral sequences for ko ∧ ( BG ) − V λ and ko ∧ ( BG ) E .Proof. The E -pages of these Adams spectral sequences are determined by the A (1)-module structures oncohomology, which are in turn determined by w and w of the virtual bundles 3 − V λ and E . Since w ( E ) = x and w ( E ) = u , then for i = 1 , w i (3 − V λ ) = w i ( E ). (cid:3) Because E is induced from a representation which is an exterior sum, its Thom spectrum splits as(5.80) ( BG ) E ’ ( BS ) − V λ | S ∧ ( B Z / σ − The Künneth theorem then simplifies the E -page:(5.81) E s,t = Ext s,t A (1) ( e H ∗ (( BS ) − V λ | S ) ⊗ Z / e H ∗ (( B Z / σ − ) , Z / . Campbell [Cam17, Figure 6.1] computes the A (1)-module structure on e H ∗ (( B Z / σ − ), and we computed e H ∗ (( BS ) − V λ ) in (5.63) (drawn in Figure 18). Proposition 5.82.
There is an isomorphism of A (1) -modules (5.83) e H ∗ (( B Z / σ − ) ⊗ Z / N ∼ = Σ N ⊕ ( V ⊗ Z / A (1)) ⊕ P , where N is as in Figure 23, V is a graded vector space with a homogeneous basis in degrees { , , , } , and P is -connected.Proof. Compute directly, by hand or by computer. (cid:3)
Recall from (5.63) (drawn in Figure 18) the A (1)-module structure on ( BS ) − V λ . Margolis’ theorem(Theorem 3.24) splits off a Σ k H Z / ko ∧ ( BS ) − V λ for every direct summand of Σ k A (1) in e H ∗ (( BS ) − V λ ); below degree 8, this occurs for k = 3 ,
5. Therefore, by the same line of reasoning as in §5.2,there is a spectrum Y such that(5.84) e ko n (( BG ) − V λ ) ∼ = π n ( Y ) ⊕ e H n − (( B Z / σ − ) ⊕ e H n − (( B Z / σ − ) , and as A -modules,(5.85) e H ∗ ( Y ) ∼ = A ⊗ A (1) ( N ⊕ Σ N ⊕ P ) ⊗ Z / e H ∗ (( B Z / σ − ) , where P is a 4-connected A (1)-module. Therefore the change-of-rings formula (3.4) applies to the E -pageof the Adams spectral sequence for π ∗ ( Y ), showing(5.86) E s,t ( Y ) ∼ = Ext s,t A (1) (( N ⊕ Σ N ⊕ P ) ⊗ Z / e H ∗ (( B Z / σ − ) , Z / . To calculate the spin bordism groups of ( BG ) − V λ , we will work with this spectral sequence, adding thesummands corresponding to Σ H Z / H Z / Theorem 5.87.
The first few spin bordism groups of ( BG ) − V λ are e Ω Spin0 (( BG ) − V λ ) ∼ = Z / e Ω Spin1 (( BG ) − V λ ) ∼ = Z / e Ω Spin2 (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin3 (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin4 (( BG ) − V λ ) ∼ = ( Z / ⊕ , and e Ω Spin5 (( BG ) − V λ ) is torsion, so the th ( S × Z / -equivariant phase homology group for this case isisomorphic to ( Z / ⊕ .Proof. Proposition 5.82 and (5.86) together imply the E -page for Y is(5.88) E s,t ( Y ) ∼ = Ext s,t A (1) (Σ N ⊕ V ⊗ Z / A (1) ⊕ Σ N ⊕ Σ V ⊗ Z / A (1) ⊕ P, Z / , where P is 4-connected. We will see that the E -page in t − s ≤ s ≥
2, so there can be nodifferentials involving Ext( P ) in the range we care about.Our first order of business is therefore to determine Ext s,t A (1) ( N , Z /
2) for small s, t . There is an extensionof A (1)-modules(5.89) 0 (cid:47) (cid:47) R (cid:47) (cid:47) N (cid:47) (cid:47) Σ R (cid:47) (cid:47) , which we draw in Figure 23, left, fitting Ext s,t A (1) ( N , Z /
2) into a long exact sequence (Figure 23, right). The A (1)-module R and its Ext are calculated in the range we need by Freed-Hopkins [FH16a, Figure 5, case s = 3] and Beaudry-Campbell [BC18, Figures 32, 33]. In the range pictured, there are two boundary maps inFigure 23, right, which could be nonzero; the existence of a nonzero map N → Σ Z / δ : Ext , ( R ) → Ext , (Σ R ) to vanish. We do not need to know whether the other pictured boundarymap vanishes. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 65 R N Σ R s ↑ t − s → Figure 23.
Left: the A (1)-module N in the extension (5.89). Right: the correspondinglong exact sequence in Ext.Hence the E -page for computing π ∗ ( Y ) is(5.90) s ↑ t − s → π ∗ ( Y ); for the each factor of e H ∗− ‘ (( B Z / σ − ), add a single Z / ‘ and above. (cid:3) Class D, spin- / case. Now we ask for the symmetries to mix. Let f D / denote the local system ofsymmetry types for this case. By Theorem 2.11, we consider the spin bordism of X := ( BS × B Z / Det( V λ ) − ,because V λ is not pin − . The isomorphism Det V λ ∼ = 0 (cid:1) σ provides an isomorphism X ’ ( BS ) + ∧ ( B Z / σ − ,so (2.10b) implies the spin bordism of this spectrum computes the pin − bordism of BS , which could beindependently interesting. Theorem 5.91.
The first few spin bordism groups of X are e Ω Spin0 ( X ) ∼ = Z / e Ω Spin1 ( X ) ∼ = ( Z / ⊕ e Ω Spin2 ( X ) ∼ = Z / ⊕ Z / ⊕ Z / e Ω Spin3 ( X ) ∼ = ( Z / ⊕ e Ω Spin4 ( X ) ∼ = ( Z / ⊕ . Since e Ω Spin5 ( X ) is torsion by Lemma 3.26, Ph S × Z / ( R ; f D / ) ∼ = Z / .Proof. As usual, Lemma 3.30 spits X as a sum of ( B Z / σ − and another spectrum M , where e H ∗ ( M ; Z / e H ∗ ( X ; Z /
2) to the space spanned by { U w ( λ ) k } . The ( B Z / σ − summand gives uspin − bordism, and we focus on M .We have w (Det( V λ ) −
1) = w ( V λ ) = x and w (Det V λ −
1) = 0; this and the A -module structure on BS × B Z / A (1)-module structure on M . We obtain an isomorphism of A (1)-modules(5.92) e H ∗ ( M ; Z / ∼ = Σ R ⊕ Σ R ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ Σ A (1) ⊕ P, where P is 4-connected. We will see momentarily that for t − s ≤ E s,t is empty for s ≥
2; this and the4-connectedness of P imply its contribution to the E -page cannot affect the spectral sequence in degrees t − s ≤
4, which is all we need. We draw these summands, except for P , in Figure 24. Ua UaxUa x Ub Uα Ua Uab Uax Uc Ua b Ubx Figure 24.
The A (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. The picturedsummand contains all classes in degrees 4 and below. Here α := b x + a b + c .Freed-Hopkins [FH16a, Figure 5, cases s = ±
3] and Beaudry-Campbell [BC18, Figures 32, 33, 37] calculateExt( R ) and Ext( R ) in the degrees we need, and we can draw the E -page of the Adams spectral sequence:(5.93) s ↑ t − s → − bordism groups from the ( B Z / σ − , which are computed in [ABP69,KT90b] split off of X sum together to the groups in the theorem. (cid:3) Class A, spinless case.
Let f A denote the local system of symmetry types for this case. We want to cal-culate e Ω Spin c ∗ (( BG ) − V λ ). By Lemma 3.22, e H ∗ (( BG ) − V λ ); Z /
2) is 2-torsion, and therefore e Ω Spin c ∗ (( BG ) − V λ )is too, so it suffices to use the 2-primary Adams spectral sequence. Theorem 5.94.
The first few spin c bordism groups of ( BG ) − V λ are: e Ω Spin c (( BG ) − V λ ) ∼ = Z / e Ω Spin c (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin c (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin c (( BG ) − V λ ) ∼ = ( Z / ⊕ e Ω Spin c (( BG ) − V λ ) ∼ = Z / ⊕ ( Z / ⊕ , and e Ω Spin c (( BG ) − V λ ) is torsion. Hence Ph S × Z / ( R ; f A ) ∼ = Z / ⊕ ( Z / ⊕ .Proof. There is an isomorphism of E (1)-modules(5.95) N ∼ = E (1) ⊕ Σ R ⊕ Σ R , hence another isomorphism of E (1)-modules(5.96) e H ∗ (( BG ) − V λ ) ∼ = ( V c ⊗ Z / A (1)) ⊕ Σ R ⊕ Σ R ⊕ P c , NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 67 where P c is 4-connected and V c is a graded vector space with a homogeneous basis of elements in degrees { , , , , , , , , , , } . Therefore the E -page of the Adams spectral sequence is(5.97) s ↑ t − s → (cid:3) Class A, spin- / case. Because V λ is not pin c , Theorem 2.24 tells us to compute the spin c bordismgroups of X := ( BS × B Z / Det( V λ − . Theorem 5.98.
The first few spin c bordism groups of X are e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = Z / e Ω Spin c ( X ) ∼ = Z / ⊕ ( Z / ⊕ e Ω Spin c ( X ) ∼ = ( Z / ⊕ e Ω Spin c ( X ) ∼ = Z / ⊕ Z / ⊕ ( Z / ⊕ . As e Ω Spin c ( X ) is torsion, the th ( S × Z / -equivariant phase homology group for this case is isomorphic to Z / ⊕ Z / ⊕ ( Z / ⊕ .Proof. By Lemma 3.30, X splits as ( B Z / σ − ∨ M , where e H ∗ ( M ; Z /
2) is isomorphic to a complementarysubspace to the subspace Z / · { U x k } inside e H ∗ ( X ; Z / B Z / σ − summand contributespin c bordism groups to the final answer, so we focus on M . The A (1)-module structure we computed in (5.92)and drew in Figure 24 tells us the E (1)-structure; here, we use that R ∼ = E (1) ⊕ Σ R and R ∼ = E (1) ⊕ Σ R as E (1)-modules. Therefore, there is an E (1)-module isomorphism(5.99) e H ∗ ( M ; Z / ∼ = Σ E (1) ⊕ Σ R ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ E (1) ⊕ Σ R ⊕ Σ E (1) ⊕ Σ E (1) ⊕ P, where P is 4-connected. Therefore to infer anything about e Ω Spin c ( M ) from this spectral sequence, we mustargue that P does not affect it; this will follow when we see the t − s = 4 line of the E -page is empty inAdams filtration 2 and above, so there can be no nonzero differentials from the 5-line to the 4-line. Wedraw (5.99) in Figure 25.Recalling Ext( R ) from Proposition 4.42, the E -page of the Adams spectral sequence for f ku ∗ ( M ) is(5.100) s ↑ t − s → e Ω Spin c ∗ ( M ) and combine it with pin c bordism ascomputed in [BG87a, BG87b] to conclude. (cid:3) Ua Uax Ub Ua Uab Uax Uc Uα Ua b Ubx Figure 25.
The E (1)-module structure on e H ∗ ( M ; Z /
2) in low degrees. Here α := abx + b + cx . This submodule contains all elements in degrees 4 and below.5.6. Chiral icosahedral symmetry.
Let λ : A → SO denote the representation given by chiral icosahedralsymmetry, and as usual let V λ → BA denote the associated vector bundle. Remark . Unlike the previous symmetry groups we studied, icosahedral symmetry is incompatible withtranslations, and there are no space groups whose underlying point group is either the chiral icosahedralgroup or the full icosahedral group. This means one should not expect to realize any phases equivariant forthese symmetry groups as a lattice Hamiltonian system on a periodic lattice on R . This does not rule out thepossibility of interesting phases with an icosahedral symmetry: there are examples of phases studied via latticeHamiltonian realizations on lattices in great generality, such as the toric code model in [Fre19, §2.3], the GDSmodel in [FH16b, Deb20, FHHT20], and the phases on aperiodic lattices studied by Huang-Wu-Liu [HWL20].In a similar vein, it may be possible for a Hamiltonian on an aperiodic lattice with icosahedral symmetry tomodel a nontrivial crystalline SPT. See [VLP +
19] for an example of how such an implementation might look.For icosahedral symmetry, the hard work is behind us. Let λ : A → O denote the representation as theorientation-preserving symmetries of the icosahedron. The restriction to A ⊂ A corresponds to symmetriesthat preserve a concentric tetrahedron. Let V λ → BA be the associated bundle to λ . Lemma 5.102.
The inclusion ϕ : A , → A induces an equivalence on mod cohomology. Hence ϕ induces -primary equivalences Σ ∞ ( BA ) + → Σ ∞ ( BA ) + and ( BA ) − ϕ ∗ ( V λ ) → ( BA ) − V λ .Proof. The first part is Lemma 3.29: here [ A : A ] = 5, P = Z / × Z /
2, and for both A and A , N ( P ) /P ∼ = Z / ϕ : ( BA ) − ϕ ∗ ( V λ ) → ( BA ) − V λ inducesan isomorphism on mod 2 cohomology. The desired 2-primary equivalences then follow from the mod 2Whitehead theorem [Ser53, Chapitre III, Théorème 3]. (cid:3) We can therefore reuse the calculations we made at the prime 2 in §5.1 to obtain the 2-primary parts of e Ω Spin k (( BA ) − V λ ) and Ω Spin k ( BA ); the odd-primary pieces are different, but not hard. Proposition 5.103.
The only odd-primary torsion in H k ( BA ) for k < is contained in H ( BA ) ∼ = Z / .Proof sketch. One can compute this using
Gap ; we also indicate how to do it by hand. Since | A | = 60 = 2 · · p -primary torsion for p >
5, so it suffices to determine H k ( BA ; Z /
3) and H k ( BA ; Z /
5) in lowdegrees. This can be done using the theorem of Adem-Milgram [AM04, Theorem II.6.8] mentioned above,since the Sylow 3- and 5-subgroups of A are abelian. (cid:3) Corollary 5.104. In e Ω Spin k (( BA ) − V λ ) and Ω Spin k ( BA ) , the only odd-primary torsion for k < is a Z / in degree .Proof. As usual, we use the fact that Ω
Spin ∗ → Ω SO ∗ is an isomorphism on odd-primary torsion, together withthe Thom isomorphism e Ω SO ∗ (( BA ) − V λ ) ∼ = Ω SO ∗ ( BA ), to reduce to showing the claim for Ω SO k ( BA ). For NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 69 this, use the Atiyah-Hirzebruch spectral sequence(5.105) E p,q = H p ( BA ; Ω SO q (pt)) = ⇒ Ω SO p + q ( BA ) . On the E -page, the only odd-primary torsion in total degree below 7 is Z / ⊂ E , = H ( BA ). In alldifferentials involving E r , , the other group is zero, so this odd-primary torsion lives to the E ∞ -page.We also must check that the free summands in total degree below 7 do not receive differentials that producemore odd-primary torsion. There are only two such free summands, in E , and E , , and they can onlyreceive differentials from 2-torsion abelian groups, so that does not happen. (cid:3) Now we need to combine this with the 2-primary summands. For ( BA ) − V λ , we need Ω Spin ∗ (( BA ) − ϕ ∗ V λ ),which we computed in Theorem 5.4. For BA , we need Ω Spin ∗ ( BA ); in the degrees we need, this is isomorphicto ko ∗ ( BA ), which Bruner-Greenlees compute in [BG10, §7.7.E]. Theorem 5.106.
The low-degree spin bordism groups of ( BA ) − V and BA are e Ω Spin0 (( BA ) − V λ ) ∼ = Z Ω Spin0 ( BA ) ∼ = Z e Ω Spin1 (( BA ) − V λ ) ∼ = 0 Ω Spin1 ( BA ) ∼ = Z / e Ω Spin2 (( BA ) − V λ ) ∼ = 0 Ω Spin2 ( BA ) ∼ = Z / ⊕ Z / e Ω Spin3 (( BA ) − V λ ) ∼ = Z /
30 Ω
Spin3 ( BA ) ∼ = Z / e Ω Spin4 (( BA ) − V λ ) ∼ = Z Ω Spin4 ( BA ) ∼ = Z e Ω Spin5 (( BA ) − V λ ) ∼ = Z / ⊕ Z / Spin5 ( BA ) ∼ = 0 e Ω Spin6 (( BA ) − V λ ) ∼ = Z / Spin6 ( BA ) ∼ = Z / . Hence the th A -equivariant phase homology groups vanish for both spinless and spin- / fermions. Finally, class A. Since V λ is not pin c , because its restriction to A is not (Lemma 5.7), we care about( BA ) Det( V λ ) − ∼ = ( BA ) + in the spin-1 / V λ is orientable. Let f A , resp. f A / , denote theequivariant local systems of symmetry types for the class A spinless, resp. spin-1 / Theorem 5.107.
The low-degree spin c bordism groups of ( BA ) − V λ and BA are e Ω Spin c (( BA ) − V λ ) ∼ = Z Ω Spin c ( BA ) ∼ = Z e Ω Spin c (( BA ) − V λ ) ∼ = 0 Ω Spin c ( BA ) ∼ = 0 e Ω Spin c (( BA ) − V λ ) ∼ = Z Ω Spin c ( BA ) ∼ = Z ⊕ Z / e Ω Spin c (( BA ) − V λ ) ∼ = Z /
30 Ω
Spin c ( BA ) ∼ = Z / e Ω Spin c (( BA ) − V λ ) ∼ = Z Ω Spin c ( BA ) ∼ = Z , and in both cases, Ω Spin c is torsion. Hence both Ph A ( R ; f A ) and Ph A ( R ; f A / ) vanish.Proof. The calculation separates into 2-primary and odd-primary computations; by Lemma 5.102, the2-primary pieces are exactly as in Theorem 5.8.The calculation of the odd-primary parts follows the same line of reasoning as the proof of Lemma 5.71:as usual, use the odd-primary equivalence
MTSpin c → MTSO ∧ ( B U ) + . We know from Proposition 5.103that the only odd-primary torsion in H k ( BA ) for k ≤ Z / ⊂ H ; feeding that to the Künnethformula, the only odd-primary torsion in H k ( B U × BA ) is two Z /
15 summands in H and H . Then theAtiyah-Hirzebruch argument is identical to the argument in Lemma 5.71. (cid:3) Full icosahedral symmetry.
If one includes orientation-reversing symmetries of the icosahedron, thesymmetry group enlarges to A × Z /
2, with the Z / A × Z / Theorem 5.108.
Let ρ be a virtual A × Z / -representation with rank zero, and let V ρ → BG denote theassociated virtual vector bundle. Suppose that w ( V ρ ) = x , where x denotes the generator of H ( B Z / Z / ⊂ H ( B ( A × Z / Z / . Then inclusion of the pyritohedral symmetry subgroup ϕ : A × Z / , → A × Z / induces a homotopy equivalence B ( A × Z / V ρ ’ → B ( A × Z / V ρ .Proof. By the Whitehead theorem, it suffices to establish that ϕ induces an isomorphism e H ∗ ( B ( A × Z / V ρ ; k ) → e H ∗ ( B ( A × Z / V ρ ; k ) for k = Q and k = Z /p for all primes p .Lemma 5.102 and the Künneth theorem imply that ϕ ∗ : H ∗ ( B ( A × Z / Z / → H ∗ ( B ( A × Z / Z / k = Z / G be either of A × Z / A × Z /
2; the map Bϕ : B ( A × Z / → B ( A × Z /
2) allows us to thinkof V ρ as over BG for either G , and make sense of the statement w ( V ρ ) = x . The Thom isomorphismimplies e H ∗ (( BG ) V ρ ; Z ) ∼ = H ∗ ( BG ; Z x ), and since Z x arises as a pullback local system along BG → B Z / e H ∗ ( BG ; Z ) is 2-torsion. The universal coefficients theorem then implies that when wetake coefficients in k = Q or k = Z /p for p odd, e H ∗ ( B ( A × Z / V ρ ; k ) and H ∗ ( B ( A × Z / V ρ ; k ) vanish, sothe map between them is vacuously an isomorphism. (cid:3) Let λ : A × Z / → O denote the representation as the group of symmetries of an icosahedron and V λ → B ( A × Z /
2) denote the associated vector bundle. Then w ( V λ ) = x . Let f D and f D / denote thespinless, resp. spin-1 / f A and f A / denote theiranalogues in class A. Corollary 5.109. ϕ induces homotopy equivalences B ( A × Z / − V λ ∼ = −→ ( B ( A × Z / − V λ (5.110a) ( B ( A × Z / Det( V λ ) − ∼ = −→ ( B ( A × Z / Det( V λ ) − . (5.110b) Therefore(1) Proposition 5.11 implies that Ph A × Z / ( R ; f D ) ∼ = ( Z / ⊕ ;(2) Theorem 5.26 implies that Ph A × Z / ( R ; f D / ) ∼ = Z / ;(3) Theorem 5.28 implies that Ph A × Z / ( R ; f A ) ∼ = Z / ⊕ ( Z / ⊕ ; and(4) Theorem 5.31 implies that Ph A × Z / ( R ; f A / ) ∼ = Z / ⊕ ( Z / ⊕ . Glide symmetry protected phases
Though we have focused on point group symmetries thus far, Freed-Hopkins’ ansatz [FH19a] also ap-plies to crystallographic groups. In this section, we apply their ansatz to the group of glide symme-tries; invertible phases equivariant for this symmetry have been studied by Lu-Shi-Lu [LSL17] and Xiong-Alexandradinata [XA18], and our results agree with theirs. In particular, Lu-Si-Lu make a conjectureclassifying certain glide-symmetric phases in all symmetry types, and we prove that their conjecture followsfrom Freed-Hopkins’ ansatz.The group of glide symmetries acting on R d , d ≥
2, is the free group on the single generator(6.1) ( x , x , . . . , x d ) ( x + 1 , − x , x , . . . , x d ) . In previous sections, when the symmetry type is H = Spin, Spin c , Pin ± , etc., the symmetry type can mixwith the group action on spacetime, corresponding physically to spinless or spin-1 / µ denotes the kernel of the map Spin n → SO n or Pin ± n → O n , all extensions(6.2) 0 (cid:47) (cid:47) µ (cid:47) (cid:47) e G (cid:47) (cid:47) Z (cid:47) (cid:47) Z -actionwith respect to mixing with fermion parity, corresponding to the trivial local system E → R d with value E := Map( MTH , Σ I Z ). Definition 6.3.
Recall from Remark 1.28 that we defined a “forgetful map” ϕ : Ph Z ∗ ( R d ; E ) → Ph ∗ ( R d ; E ).The intrinsically Z -equivariant phase homology , denoted c Ph Z ∗ ( R d ; E ), is the kernel of this map. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 71
This corresponds under Freed-Hopkins’ ansatz to what Lu-Shi-Lu call a glide SPT : an invertible phaseequivariant for a Z glide symmetry which is trivializable when one forgets the symmetry.Let TP d ( H ) denote the abelian group of SPT phases in (spatial) dimension d ; Freed-Hopkins’ ansatz [FH16a]classifying these phases in terms of invertible field theories predicts TP d ( H ) ∼ = E − d .Lu-Shi-Lu [LSL17] studied groups of glide SPTs and conjectured a formula classifying them in terms ofthe classification of ordinary SPTs. We prove the corresponding statement on phase homology groups. Theorem 6.4.
For a given symmetry type ρ n : H n → O n , there is a natural isomorphism c Ph Z ( R d ; E ) ∼ = E − ( d − ⊗ Z / . Passing this through the ansatz, this predicts that the group of glide SPTs is naturally isomorphic to TP d − ( H ) ⊗ Z /
2, which is Lu-Shi-Lu’s original conjecture [LSL17, Conjecture 1]. Xiong-Alexandradinata [XA18]also obtain this result using physics-based arguments.
Proof of Theorem 6.4.
We calculate the 0 th Z -equivariant Borel-Moore E -homology of R d . As the Z -action isfree, this is the 0 th (nonequivariant) Borel-Moore E -homology of the fundamental domain X := R d / Z . Sincethe one-point compactification X of X is a finite CW complex, this Borel-Moore homology is isomorphic to e E ( X ).If σ → S denotes the Möbius bundle, then X is diffeomorphic to the total space of σ ⊕ R d − → S , so X is the Thom space ( S ) σ + d − . The identification ( S ) σ ∼ = RP induces X ∼ = Σ d − RP , and therefore(6.5) Ph Z ∗ ( R d ; E ) ∼ = e E (Σ d − RP ) ∼ = e E − d ( RP ) . Lemma 6.6.
Let p : S → RP be the double cover map and s : e E k ( S ) → e E k +1 ( S ) be the suspensionisomorphism. The composition p ∗ ◦ δ ◦ s : e E − ( S ) → e E − ( S ) is multiplication by .Proof. This follows because the suspension is the cofiber of the cofiber; then one explicitly checks whathappens on mapping cylinders. (cid:3)
Lemma 6.7.
Under these isomorphisms, the forgetful map Ph Z ( R d ; E ) → Ph ( R d ; E ) is identified with δ .Proof. Because Z acts freely on R d , E Z , BM ( R d ) is identified with e E of the one-point compactification of R d / Z , which we saw above is homeomorphic to Σ d − RP . The codomain of the forgetful map is E , BM ( R d ) ∼ = e E (Σ d − S ), so we have identified δ with a map e E (Σ d − RP ) → e E (Σ d − S ). But tracing through theconstruction in Remark 1.28, this map comes from applying e E to an actual map Σ d − RP → Σ d − S .Next, precompose with Σ d +2 p : Σ d − S → Σ d − RP and check that this map has degree 2, agreeing withLemma 6.6. This suffices to identify the maps because p ∗ : [ RP , S ] → [ S , S ] is injective. (cid:3) RP is homeomorphic to the cofiber of a degree-2 map S → S . Hence there is a long exact sequence inreduced E -homology(6.8) · · · (cid:47) (cid:47) e E − d ( S ) m (cid:47) (cid:47) e E − d ( S ) r (cid:47) (cid:47) e E − d ( RP ) δ (cid:47) (cid:47) e E − d ( S ) (cid:47) (cid:47) · · · where m is multiplication by 2. Exactness implies ker( δ ) = Im( r ) = coker( m ). Using the suspensionisomorphism, e E k ( S ) ∼ = e E k − , and therefore coker( m ) ∼ = E − ( d − ⊗ Z /
2, and 6.7 identifies δ with the forgetfulmap from equivariant to nonequivariant phase homology for R d . In particular, c Ph Z ( R d ; E ) ∼ = ker( δ ), whichwe have naturally identified with E − ( d − ⊗ Z / (cid:3) Remark . Using the long exact sequence (6.8), we observe that Ph Z ( R d ; E ) has exponent 4. This is becausefor any long exact sequence of abelian groups(6.10) · · · (cid:47) (cid:47) A · (cid:47) (cid:47) A f (cid:47) (cid:47) B g (cid:47) (cid:47) C · (cid:47) (cid:47) C (cid:47) (cid:47) . . . in which A and C are finitely generated, Im( f ) ∼ = A/
2, hence has exponent 2, and ker( g ) is isomorphic to thesubgroup of order-2 elements of C , which also has exponent 2. Since B is an extension of ker( g ) by Im( f ), B has exponent 4.Passing this observation through Freed-Hopkins’ ansatz, this recovers an observation of Xiong-Alexandradinata [XA18]:that any phase equivariant with respect to glide symmetry, whether a glide SPT or not, has order dividing 4. Example 6.11.
In Altland-Zirnbauer class AII, corresponding to the symmetry type pin ˜ c + , the ansatzpredicts a unique nontrivial glide SPT in dimension 2 + 1, coming from the classification(6.12) [ MTPin ˜ c + , Σ I Z ] ⊗ Z / ∼ = Z / MTPin ˜ c + , Σ I Z ] is due to Freed-Hopkins [FH16a, §9.3]). Physicists are particularlyinterested in this nontrivial glide SPT phase, which is predicted to have unusual surface states called“hourglass fermions” [WACB16], and which has been studied experimentally [MYL + Conclusion and outlook
We conclude by indicating a few directions of potential further research.7.1.
From free fermions to interacting phases.
Free fermion phases are a rich source of examples ofinvertible phases in the physics literature, at least for symmetry types spin, pin ± , spin c , etc. The classificationof free fermion systems uses K -theory: see Kitaev [Kit09] for the original proposal and Freed-Moore [FM13]for a comprehensive classification. However, for a given dimension and symmetry type, the map fromfree fermion systems to invertible phases of matter can in general have both kernel (as first observed byFidkowski-Kitaev [FK10, FK11] and Turner-Pollmann-Berg [TPB11]) and cokernel (as first observed byWang-Potter-Senthil [WPS14] and Wang-Senthil [WS14]). Researchers are also interested in the free-to-interacting map for phases with spatial symmetries, and this map has been studied from a physics point ofview for crystalline phases in several works, including [YR13, IF15, MFM15, LTH16, SS17, RL18, Zou18,LVK19, ZYQG20, ACR + MTSpin → KO [ABS64], but they do not consider spatial symmetries. In viewof the large bodies of research on free fermions with spatial symmetries and invertible phases with spatialsymmetries, it would be nice to understand the map between them in the presence of spatial symmetry from thelow-energy field theory perspective, and to make contact with the work of Adem, Antolín Camarena, Semenoff,and Sheinbaum [AACSS16], Sheinbaum and Antolín Camarena [SC20], and Cornfeld-Carmeli [CC21] studyingfree fermion phases with spatial symmetries using methods from homotopy theory. This is something wehope to tackle in future work.7.2. Other symmetry types.
We investigated two of the ten Altland-Zirnbauer classes, and it would beinteresting to know whether a version of the FCEP holds for other classes. One starting point could be classC, corresponding to a spin h structure [FH16a, (9.25)]; the calculations in §2.8 could be applied to Spin hn toobtain a fermionic crystalline equivalence principle for class C and hopefully phase homology calculationspredicting the existence of additional crystalline SPT phases.Several teams of researchers have studied or classified interacting fermionic crystalline SPTs for otherAltland-Zirnbauer types, including [YR13, YX14, CHMR15, LTH16, WF17, CW18, RL18, SXG18, MSH19,ZXXS20, ZYQG20]. It would be good to compare their computations with the predictions made by an FCEPin other symmetry types.Another interesting potential connection with preexisting work is the case of class A phases with a spatialreflection interacting with the internal U symmetry. Depending on how the symmetries mix, Shiozaki-Shapourian-Gomi-Ryu [SSGR18, §V.C, §V.E] and Thorngren-Else [TE18, §VII.B] obtain classifications interms of pin ˜ c ± bordism, and we would be interested in knowing whether that can also be obtained from ouransatz. Similarly, can one begin with class C phases and a reflection acting on the internal SU symmetryand obtain a classification in terms of pin h ± bordism?7.3. Crystallographic groups.
Though we discussed glide symmetries in §6, we have barely touched uponthe rich world of crystallographic groups. Free-fermion phases equivariant for these groups have been studied,but much less is known about the interacting case, even though the our ansatz applies to it. There are someclassifications by other methods for various classes of crystallographic groups; for example, Ouyang-Wang-Gu-Qi [OWGQ20] study wallpaper group symmetries, and Sheinbaum-Antolín Camarena [SC20] provide a generalframework and a few examples. There is also work by Wang-Alexandradinata-Cava-Bernevig [WACB16] and Spin h is the symmetry type Spin × µ SU → O. Freed-Hopkins [FH16a, Proposition 9.16] call this symmetry type G ; itis sometimes also called spin-SU , e.g. in [WWW19]. Likewise, the symmetry types pin h ± we refer to later in this section aredefined to be Pin ± × µ SU , and are called G ± by Freed-Hopkins [FH16a, Proposition 9.16]. NVERTIBLE PHASES FOR MIXED SPATIAL SYMMETRIES 73
Guo-Ohmori-Putrov-Wan-Wang [GOP +
20] studying interacting phases for specific crystallographic groupsthat are not point groups.7.4.
Lattice realizations.
Modeling topological phases as lattice Hamiltonian systems can make anycrystallographic symmetries acting on space very explicit, using a lattice and Hamiltonian invariant underthe symmetry of interest. Our predictions of point group SPTs should correspond to actual lattice models ofphases. We listed several specific predicted phases of interest in §3.1, and these would make for good startingpoints for lattice realizations.
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