Stable diffeomorphism classification of some unorientable 4-manifolds
aa r X i v : . [ m a t h . G T ] F e b STABLE DIFFEOMORPHISM CLASSIFICATION OF SOME UNORIENTABLE4-MANIFOLDS
ARUN DEBRAY
Abstract.
Kreck’s modified surgery theory reduces the classification of closed, connected 4-manifolds, upto connect sum with some number of copies of S × S , to a series of bordism questions. We implementthis in the case of unorientable 4-manifolds M and show that for some choices of fundamental groups, thecomputations simplify considerably. We use this to completely solve the case when π ( M ) is finite of order2 mod 4: for each such choice of π ( M ), there are nine stable diffeomorphism classes for which M is pin + ,one stable diffeomorphism class for which M is pin − , and four stable diffeomorphism classes for which M isneither. We also determine the corresponding stable homeomorphism classes. Introduction
The classification of closed 4-manifolds up to diffeomorphism is impossible in general: a solution wouldalso solve the word problem for groups. Even if one fixes the fundamental group to avoid this problem,the classification is still currently intractable. For this reason, topologists study weaker classifications of4-manifolds which are coarse enough to be calcuable yet fine enough to be useful.Stable diffeomorphism is an example of such an invariant. Two closed 4-manifolds M and N are stablydiffeomorphic if there are m, n ≥ M m ( S × S ) is diffeomorphic to N n ( S × S ). This notionof equivalence has applications to quantum topology: for example, Reutter [Reu20, Theorem A] shows thatthe partition functions of 4d semisimple oriented TFTs are insensitive to stable diffeomorphism along the wayto showing that such TFTs cannot distinguish homotopy-equivalent closed, oriented 4-manifolds. And stablediffeomorphism classes are computable: once the fundamental group G is fixed, Kreck [Kre99] shows how toreduce the classification of 4-manifolds up to stable diffeomorphism to a collection of bordism computations,and for many choices of G , the classification of closed, connected, oriented 4-manifolds with π ( M ) ∼ = G upto stable diffeomorphism has been completely worked out, thanks to work of Wall [Wal64], Teichner [Tei92],Spaggiari [Spa03], Crowley-Sixt [CS11], Politarczyk [Pol13], Kasprowski-Land-Powell-Teichner [KLPT17],Pedrotti [Ped17], Hambleton-Hildum [HH19], and Kasprowski-Powell-Teichner [KPT20].Researchers interested in topological manifolds also study stable homeomorphism of topological manifolds,i.e. homeomorphism after connect-summing with some number of copies of S × S . Kreck’s theorem appliesto this case too, reframing the question in terms of bordism of topological manifolds. Stable homeomor-phism classifications are studied by Teichner [Tei92, §5], Wang [Wan95], Hambleton-Kreck-Teichner [HKT09],Kasprowski-Land-Powell-Teichner [KLPT17, §§4–5], Hambleton-Hildum [HH19], and Kasprowski-Powell-Teichner [KPT20, §2.3],Much less work has been done on unorientable 4-manifolds, even though the theory still works and is insome respects simpler, as we explain below. There is some work in the literature, such as that of Kreck [Kre84],Wang [Wan95], Kurazono [Kur01], Davis [Dav05], and Friedl-Nagel-Orson-Powell [FNOP19, §12].The goal of this paper is to compute sets of stable diffeomorphism and stable homeomorphism classesfor a large class of unorientable 4-manifolds, as well as determining the corresponding complete stablediffeomorphism and homeomorphism invariants. As a consequence of our Theorem 2.1, for many finitegroups G , the classification of stable diffeomorphism or homeomorphism classes of unorientable 4-manifoldswith π ( M ) ∼ = G reduces to the stable classifications for a smaller 2-group. For example, we show thatthe stable diffeomorphism, resp. homeomorphism classification when π ( M ) ∼ = Z / G of order 2 mod 4. We then compute theseclassifications using Kreck’s techniques. Theorem (Main theorem) . Let G be a finite group of order . Date : February 9, 2021. (1)
There are fourteen equivalence classes of closed, connected, unorientable -manifolds M up to stablediffeomorphism: nine for which M is pin + , one for which M is pin − , and four for which M isneither. (2) There are twenty equivalence classes of closed, connected, unorientable topological -manifolds M upto stable homeomorphism: ten for which M is pin + , two for which M is pin − , and eight for which M is neither. This is a combination of Theorems 3.1, 3.5, 4.2, and 4.5. In those theorems we also determine completestable diffeomorphism/homeomorphism invariants for these manifolds. The classification for M neither pin + or pin − can be extracted from work of Davis [Dav05, Theorem 2.3], but the other parts are new.We prove these theorems by establishing isomorphisms of bordism groups. Specifically, Kreck’s modifiedsurgery theory associates to G a set of symmetry types ξ : B → B O and expresses the set of stable diffeomor-phism classes in terms of the bordism groups Ω ξ ; we show that when | G | ≡ + , and pin − bordism.In the smooth case, the bordism groups Ω O4 , Ω Pin + , and Ω Pin − are well-known. The topological versionsof these bordism groups are less well-known, but Kirby-Taylor [KT90b, §9] compute Ω TopPin ± and provideenough information for us to compute Ω Top4 , which we do in Proposition 4.7.The argument we use to establish the isomorphism from ξ -bordism to a simpler kind of bordism appliesto more general choices of π ( M ). Theorem 2.1.
Suppose G is a finite group fitting into an extension (0.1) 1 / / H / / G ϕ / / P / / , where | H | is odd and P is a -group. For any unorientable virtual vector bundle V → BP , ϕ induces anequivalence of Thom spectra ( BG ) ϕ ∗ V ≃ → ( BP ) V . The Pontrjagin-Thom construction turns this equivalence into isomorphisms of bordism groups from theunorientable symmetry types Kreck associates to G to the unorientable symmetry types for P , which wecan use to compute stable diffeomorphism classes. The proof strongly requires the assumption that V isunorientable; nothing like this is true in the oriented case.Our main theorem above solves the case | G | ≡ P ∼ = Z / × Z / Z /
4, which would suffice for many groups G of order 4 mod 8. For these choices of P , many of the neededbordism groups have already been computed in the literature for other applications. For P ∼ = Z /
4, seeBotvinnik-Gilkey [BG97, §5]; for P ∼ = Z / × Z /
2, see work of Guo-Ohmori-Putrov-Wan-Wang [GOP +
20, §7],the author in [KPMT20, Appendix F] and [Deb21, §4.4], and Wan-Wang-Zheng [WWZ20, Appendix A].We begin in §1 with a quick review of Kreck’s theorem [Kre99] on stable diffeomorphism classes of 4-manifolds within a given 1-type. In §2, we study the Thom spectra of unorientable vector bundles over BG , where G is a finite group, proving Theorem 2.1. In §3, we specialize to the case where | G | ≡ RP is homeomorphic but not stably diffeomorphic to Cappell-Shaneson’s fake RP . This fact was known to Cappell-Shaneson [CS71, CS76] and the proof using Kreck’ssurgery theory is due to Stolz [Sto88]. In §4, we consider stable homeomorphism classes of topologicalmanifolds with | π ( M ) | ≡ Acknowledgments.
I thank my advisor, Dan Freed, for his constant help and guidance. I also would liketo thank Matthias Kreck and Riccardo Pedrotti for some helpful conversations related to this paper.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.
Review: normal -types, normal -smoothings, and stable diffeomorphism classes We review some standard definitions in this area. We will always assume our manifolds are closed andconnected. Except in §4, we also assume they are smooth.
TABLE DIFFEOMORPHISM CLASSIFICATION OF SOME UNORIENTABLE 4-MANIFOLDS 3
Definition 1.1. A normal -type of a manifold M is a fibration ξ : B → B O such that there is a lift of themap ν : M → BO classifying the stable normal bundle of M to a map e ν : M → B such that ξ ◦ e ν = ν , e ν is2-connected, and ξ is 2-coconnected.A choice of such a lift is called a normal -smoothing of M .Any two normal 1-types of a given manifold are homotopy equivalent as spaces over B O, so we will abusenotation and say “the” normal 1-type.The map ξ : B → B O determines a bordism theory of manifolds with a lift of the stable normal bundleacross ξ , which we denote Ω ξ ∗ ; a normal 1-smoothing of M determines a class in this bordism group. Differentnormal smoothings of the same manifold do not always define the same class in Ω ξ ∗ .Let V SO → B SO, V Spin → B Spin, etc., denote the tautological stable vector bundles over their respectivespaces. We use the convention that maps to B O are represented by rank-zero virtual vector bundles, whichis why we write E − dim E in (1.3), for example. Example 1.2 (Kreck [Kre99, §2, Proposition 2]) . When M is unorientable, Kreck classifies the possiblenormal 1-types of M into two families. Let M ′ → M be the universal cover of M , which is classified by amap θ : M → Bπ ( M ). Almost spin: If M ′ admits a spin structure, M is called almost spin . In this case, w ( M ) = θ ∗ x and w ( M ) = θ ∗ x for some x , x ∈ H ∗ ( BG ; Z / E → BG such that w i ( E ) = x i for i = 1 , Then, the normal 1-type of M is(1.3) B Spin × Bπ ( M ) V Spin ⊕ ( E − dim E ) (cid:15) (cid:15) M ν / / ♣♣♣♣♣♣♣♣♣♣♣♣ B O . Totally non-spin: If M ′ does not admit a spin structure, M is called totally non-spin . In this case, w ( M ) = θ ∗ x for some x ∈ H ( BG ; Z / E → BG be a line bundle with w ( E ) = x . Then thenormal 1-type of M is(1.4) B SO × Bπ ( M ) V SO ⊕ ( E − (cid:15) (cid:15) M ν / / ♣♣♣♣♣♣♣♣♣♣♣ B O . Because S × S has trivial stable normal bundle, taking connect sum with S × S does not changethe normal 1-type of a 4-manifold; thus the classification of 4-manifolds up to stable diffeomorphism canproceed one normal 1-type at a time. Moreover, because S × S is null-bordant, one might conclude thatstably diffeomorphic 4-manifolds M and N are bordant — or, more precisely, that M and N admit normal1-smoothings which are bordant in Ω ξ . So a plausible lower bound for the set of stable diffeomorphismclasses with normal 1-type ξ would be Ω ξ modulo some identifications arising from inequivalent normal1-smoothings of the same underlying manifold. Remarkably, this turns out to be a complete classification! Theorem 1.5 (Kreck [Kre99, Theorem C; §3, Proposition 4]) . (1) If M and N are -manifolds of the same normal -type ξ admitting normal -smoothings which arebordant in Ω ξ , then M is stably diffeomorphic to N . (2) If π ( ξ ) is finite, every class in Ω ξ can be realized as the normal -smoothing of a -manifold withnormal -type ξ .The upshot is that if Aut( ξ ) denotes the group of fiber homotopy equivalences of ξ → B O , the set of stablediffeomorphism classes of -manifolds with normal -type ξ is Ω ξ / Aut( ξ ) . The set of bordism classes of normal 1-smoothings of a given 4-manifold is contained within an Aut( ξ )-orbitof Ω ξ , so one effect of the quotient is to identify these as all coming from the same manifold. This will be true for all cases we consider in this paper, but is not true in general.
ARUN DEBRAY
This illustrates the standard way to calculate stable diffeomorphism classes: determine Ω ξ , then determinethe Aut( ξ )-action. These bordism groups are the homotopy groups of the Thom spectrum M ξ of ξ , so inthe next section we begin the calculation of stable diffeomorphism classes by simplifying M ξ .2.
Simplifying Thom spectra
Theorem 1.5 tells us to investigate the Thom spectra of the normal 1-types in Example 1.2. In both cases,the vector bundle is an exterior direct sum, so the Thom spectra split, as
MSpin ∧ ( Bπ ( M )) V in the almostspin case and MSO ∧ ( BG ) V in the totally non-spin case, where V is a rank-zero unoriented virtual vectorbundle. We attack the problem by simplifying ( Bπ ( M )) V for some choices of π ( M ). Theorem 2.1.
Suppose G is a finite group fitting into an extension (2.2) 1 / / H / / G ϕ / / P / / , where | H | is odd and P is a -group. For any unorientable virtual vector bundle V → BP , ϕ induces anequivalence of Thom spectra ( BG ) ϕ ∗ V ≃ → ( BP ) V . We’ll prove this in a series of lemmas.
Definition 2.3.
Let K be a group, A be an abelian group, and α ∈ H ( BK ; Z / H ( BK ; Z / ∼ = Hom( K, Z / A α be the Z [ K ]-module which is the abelian group Z with the K -actionin which g ∈ K acts by ( − α ( g ) . Lemma 2.4.
With notation as in Theorem 2.1, both e H ∗ (( BG ) ϕ ∗ V ) and e H ∗ (( BP ) V ) are -torsion.Proof. Using Definition 2.3, we define the Z [ P ]-module A w ( V ) and the Z [ G ]-module A w ( ϕ ∗ P ) , which isisomorphic to the pullback of A w ( V ) by ϕ . The Thom isomorphism provides isomorphisms of gradedabelian groups H ∗ ( BP ; Z w ( V ) ) ∼ = −→ e H ∗ (( BP ) V )(2.5a) H ∗ ( BG ; Z w ( ϕ ∗ V ) ) ∼ = −→ e H ∗ (( BG ) ϕ ∗ V ) , (2.5b)so we will prove the lemma using group cohomology – specifically, using the Lyndon-Hochschild-Serre spectralsequence(2.6) E p,q = H p ( BP ; ( H q ( BH ; Z )) w ( V ) ) = ⇒ H p + q ( BG ; Z w ( ϕ ∗ V ) ) . Since E p,q ∼ = H p ( BP ; M q ) for some Z [ P ]-module M q , E p,q is 2-torsion for p > When p = 0,(2.7) E ,q ∼ = H ( BP ; H q ( BH ) w ( V ) ) ∼ = ( H q ( BH ) w ( V ) ) P . We will show this vanishes. First, H q ( BH ) is Z for q = 0 and is odd-primary torsion for q > ∤ H ). Therefore if a ∈ H q ( BH ) and − a = a , a = 0. Since w ( V ) = 0, there is some g ∈ P which acts on Z w ( V ) as −
1, hence also acts on H q ( BH ) w ( V ) as −
1, so the subgroup of invariants of H q ( BH ) w ( V ) is { } .Considering the line q = 0 proves H ∗ ( BP ; Z w ( V ) ) is 2-torsion. For H ∗ ( BG ; Z w ( ϕ ∗ V ) ), we have shownthe E -page is 2-torsion, so the graded abelian group the spectral sequence converges to is also 2-torsion. (cid:3) Lemma 2.8.
With G and P as in Theorem 2.1, ϕ ∗ : H ∗ ( BP ; Z / → H ∗ ( BG ; Z / is an isomorphism ofgraded rings.Proof. Since H has odd order, its mod 2 cohomology is Z / BH → BG → BP induced by (2.2). (cid:3) We use Maschke’s theorem as follows: if G is a finite group and k is a field of characteristic 0 or characteristic ℓ ∤ G ,the category of k [ G ]-modules is semisimple. Therefore all positive-degree Ext groups vanish, in particular H m ( BG ; M ) ∼ =Ext mk [ G ] ( Z , M ) for any k [ G ]-module M and m >
1. Combined with the universal coefficient theorem, this implies that for any Z [ G ]-module M and m > H m ( G ; M ) is torsion ( k = Q ), and lacks ℓ -torsion if ℓ ∤ G . TABLE DIFFEOMORPHISM CLASSIFICATION OF SOME UNORIENTABLE 4-MANIFOLDS 5
Proof of Theorem 2.1.
Use the homology Whitehead theorem: if f : X → Y is a map of bounded-belowspectra which induces an isomorphism on rational cohomology and on mod p cohomology for every prime p ,then f is a homotopy equivalence. Lemma 2.4 and the universal coefficient theorem imply that if k = Q or k = Z /p for an odd prime p , e H ∗ (( BG ) ϕ ∗ V ; k ) and e H ∗ (( BP ) V ; k ) both vanish, so the map between them isvacuously an isomorphism. The sole remaining case is p = 2. Since 1 ≡ − Z / w ( V ) carries thetrivial P -action; thus, the Thom isomorphism has the form(2.9a) H ∗ ( BP ; Z / ∼ = −→ e H ∗ (( BP ) V ; Z / . Analogously, there is a Thom isomorphism(2.9b) H ∗ ( BG ; Z / ∼ = −→ e H ∗ (( BG ) ϕ ∗ V ; Z / . As the Thom isomorphism is functorial with respect to pullbacks of vector bundles, Lemma 2.8 lifts to implythat(2.10) ϕ ∗ : e H ∗ (( BP ) V ; Z / −→ e H ∗ (( BG ) ϕ ∗ V ; Z / (cid:3) The case | π ( X ) | ≡ M is an unorientable manifold, the description of loops as orientation-preserving or orientation-reversingdefines a surjection p : π ( M ) → Z /
2, so π ( M ) cannot have odd order. Thus the simplest case occurs when | π ( M ) | ≡ | ker( p ) | is odd and Theorem 2.1 applies to show that if V → B Z / Bπ ( M )) p ∗ V ≃ → ( B Z / V is an equivalence.Let σ → B Z / x := w ( σ ) ∈ H ( B Z / Z / H ∗ ( B Z / Z / ∼ = Z / x ]. Because ker( p ) has odd order, the Leray-Hirsch theorem implies p ∗ : H ∗ ( B Z / Z / → H ∗ ( Bπ ( M ); Z / The almost spin case.
Example 1.2 shows there are two unorientable normal 1-types in this case: w ( ν ) = 0, so it must be the pullback of p ∗ x ∈ H ( Bπ ( M ); Z / w , we have two choices: w = 0(the normal bundle is pin + ) and w = p ∗ x (the normal bundle is pin − ).Recall that for a manifold M , M is pin ± (i.e. the tangent bundle is pin ± ) iff the normal bundle is pin ∓ .A (tangential) pin + M has a Z / η -invariant of a twisted Diracoperator [Sto88, §4]; let η ′ be the invariant assigning to a pin + M the image of this η -invariant inthe nine-element set ( Z / / ( x ∼ − x ). We will see in the proof of Theorem 3.1 that all pin + structures on M give the same value of η ′ , so we may define it as an invariant of manifolds which admit a pin + structure,without choosing such a structure. Theorem 3.1.
Fix G to be a finite group of order . There are nine stable diffeomorphism classesof unorientable -manifolds with π ( M ) ∼ = G that admit a (tangential) pin + structure, and there is a singlestable diffeomorphism class of manifolds with π ( M ) ∼ = G that admit a (tangential) pin − structure. In thepin + case, η ′ is a complete stable diffeomorphism invariant.Proof. Both choices of ( w , w ) arise from vector bundles: ( p ∗ x,
0) from p ∗ σ , and ( p ∗ x, p ∗ x ) from p ∗ (3 σ ).Thus the normal 1-types are V Spin ⊕ ( p ∗ σ −
1) : B Spin × Bπ ( M ) −→ B O(3.2a) V Spin ⊕ ( p ∗ (3 σ ) −
3) : B Spin × Bπ ( M ) −→ B O , (3.2b)and their Thom spectra are MSpin ∧ ( Bπ ( M )) p ∗ σ − , resp. MSpin ∧ ( Bπ ( M )) p ∗ (3 σ ) − . By Theorem 2.1,these are equivalent to MSpin ∧ ( B Z / σ − , resp. MSpin ∧ ( B Z / σ − . Theorem 3.3 (Peterson [Pet68, §7], Kirby-Taylor [KT90a, Lemma 6]) . There are equivalences MSpin ∧ ( B Z / σ − ≃ MTPin − and MSpin ∧ ( B Z / σ − ≃ MTPin + . These bordism groups are known. There is an important subtlety in the names of these spectra in the literature:
MPin ± denotes the Thom spectra classifyingpin ± structures on the stable normal bundle, and MTPin ± denotes the Thom spectra classifying pin ± structures on the stabletangent bundle. There are equivalences MPin ± ≃ MTPin ∓ . Information on pin ± bordism is usually stated in terms of MTPin ± . ARUN DEBRAY • In the case w ( ν ) = 0, Ω ξ ∼ = Ω Pin − ∼ = 0 [ABP69, KT90b] — all 4-manifolds with this normal 1-typeare stably diffeomorphic. • When w ( ν ) = p ∗ x , Ω ξ ∼ = Ω Pin + ∼ = Z /
16 [Gia73, KT90a, KT90b].In the latter case we have to determine the Aut( ξ )-action. RP admits two pin + structures, and Kirby-Taylor [KT90b, Theorem 5.2] choose an isomorphism Ω Pin + ∼ = → Z /
16 sending these two pin + structures to ±
1. Therefore for any equivalence class x ∈ Ω Pin + and any g ∈ Aut(Pin + ), g · x = ± x , because we canrepresent x as a disjoint union of copies of RP with some pin + structure, and the Aut(Pin + )-orbit of the RP s is {± } . The isomorphism from ξ -bordism to pin + bordism allows us to also deduce that the Aut( ξ )-orbit of a class [ M ] in Ω ξ is {± [ M ] } . We obtain nine equivalence classes: 0 , ± , ± , . . . , ± ,
8, detected bythe image of the η -invariant in ( Z / / ( x ∼ − x ). (cid:3) As a consequence of Kreck’s classification in Example 1.2, we have seen that all unorientable, almost spin4-manifolds M with π ( M ) of order 2 mod 4 are either pin + or pin − , and that this determines their normal1-type. This is not true for more general G . Example 3.4.
Cappell-Shaneson [CS71, CS76] construct a closed, smooth 4-manifold Q that is homeomor-phic but not diffeomorphic to RP , and show that Q and RP are not stably diffeomorphic. Stolz [Sto88]gives another proof of this fact by computing the classes of RP and Q in Ω ξ / Aut( ξ ). We briefly summarizeStolz’ proof.Since π ( RP ) ∼ = Z / w ( RP ) = 0, the proof of Theorem 3.1 shows M ξ ≃ MTPin + , Ω ξ ∼ = Z / ξ / Aut( ξ ) ∼ = ( Z / / ( x ∼ − x ). Stolz [Sto88] chooses anisomorphism Ω ξ ∼ = → Z /
16 and shows that it sends the two pin + structures on RP to ± + structures on Q to ±
9; therefore RP and Q are not stably diffeomorphic.3.2. The totally non-spin case.Theorem 3.5.
Fix G to be a finite group of order . There are four stable diffeomorphism classes ofunorientable, totally non-spin -manifolds with π ( M ) ∼ = G . The Stiefel-Whitney numbers w and w detectthese classes. This theorem can also be extracted from work of Davis [Dav05, Theorem 2.3], who computes a differentset of invariants.
Proof.
Example 1.2 shows there is only one unorientable normal 1-type in this case: w ( ν ) = 0, so it mustbe pulled back from p ∗ x ∈ H ( Bπ ( M ); Z / p ∗ x = w ( p ∗ σ ), the normal 1-type is(3.6) V SO ⊕ ( p ∗ σ −
1) : B SO × Bπ ( M ) −→ B Oand its Thom spectrum is
MSO ∧ ( Bπ ( M )) p ∗ σ − , which by Theorem 2.1 is equivalent to MSO ∧ ( B Z / σ − . Lemma 3.7 (Gray [Gra80, §2]) . There is an equivalence MSO ∧ ( B Z / σ − ≃ MO.
So Ω ξ ∼ = Ω O4 , and Ω O4 ∼ = Z / ⊕ Z / ξ )-action istrivial. To see this, first observe that Aut(id : B O → B O) is trivial, hence acts trivially on Ω O4 . Thus theAut( ξ )-orbit of a class in Ω ξ maps to a single class in Ω O4 , so Aut( ξ )-orbits are singletons. Therefore anycomplete bordism invariant for Ω O4 also is a complete stable diffeomorphism invariant for the normal 1-type ξ , such as ( w , w ). (cid:3) Remark . If M is pin + or pin − , then its double cover is spin, and hence M is almost spin. So totally non-spin manifolds are neither pin + nor pin − . Therefore the three normal 1-types that occur when | π ( M ) | ≡ M is unorientable are the cases pin + , pin − , and neither pin + nor pin − .4. Stable homeomorphism classes
In order to classify stable homeomorphism classes of topological 4-manifolds, we run the same story,replacing B O with B Top, where Top n is the topological group of homeomorphisms R n → R n that fix theorigin and Top := lim −→ n Top n . Given a topological manifold M , there is a map ν : M → B Top called the stable topological normal bundle , so we can define normal 1-types, and Kreck’s classification argument stillapplies in the topological setting, this time determining stable homeomorphism classes.
TABLE DIFFEOMORPHISM CLASSIFICATION OF SOME UNORIENTABLE 4-MANIFOLDS 7
Lemma 4.1.
Let M be a closed, unorientable -manifold. The possible normal -types of M are the sameas in Example 1.2, except replacing B O with B Top , B SO with B STop , and B Spin with B TopSpin .Proof.
The proof is very similar to Kasprowski-Land-Powell-Teichner’s determination of the possible normal1-types of topological 4-manifolds in the orientable case [KLPT17, Proposition 4.1]. Since the Stiefel-Whitneyclasses of a manifold are homotopy invariants, notions of almost spin and totally non-spin make sense fortopological manifolds. In the almost-spin case, we have to check that a lift M → B TopSpin × Bπ ( M )is 2-connected: the proof is the same as in the smooth case, because π ( B TopSpin) = 0. For the totallynon-spin case, π ( B STop) ∼ = Z /
2, detected by w , and since M is totally non-spin, w ( M ) = 0, so the lift issurjective on π just as in the smooth case. (cid:3) Our arguments below make use of the fact that bordism groups of topological manifolds are homotopygroups of Thom spectra, which requires a transversality argument. In dimension 4, Scharlemann [Sch76]proves the topological transversality theorem that we need. See Teichner [Tei93, §IV] for more information.Let E denote Freedman’s E manifold [Fre82]. The obstruction to admitting a triangulation defines abordism invariant Ω Top4 → Z / E .4.1. The almost spin case.
There are topological versions of spin and pin ± structures; see Kirby-Taylor [KT90b,§9] for details. Kirby-Taylor also produce a homomorphism S : Ω TopPin + → Ω TopPin − ∼ = Ω Pin − ∼ = Z / + topological 4-manifold M to the pin − bordism class of a continuously embedded representative ofthe Poincaré dual of w ( M ) , which has an induced pin − structure and a unique smooth structure. Let S ′ be the invariant sending a topological pin + M to the image of S ( M ) in the set ( Z / / ( x ∼ − x ). Theorem 4.2.
Fix G to be a finite group of order . (1) There are ten stable homeomorphism classes of unorientable pin + topological -manifolds with π ( M ) ∼ = G . These classes are detected by the invariant S ′ constructed above and the triangulation obstruction. (2) There are two stable homeomorphism classes of unorientable pin − topological -manifolds with π ( M ) ∼ = G . These classes are detected by the triangulation obstruction.Proof. Following the same line of argument as in the proof of Theorem 3.1, the two normal 1-types’ Thomspectra are
MTopSpin ∧ ( Bπ ( M )) p ∗ σ − and MTopSpin ∧ ( Bπ ( M )) p ∗ (3 σ ) − , and Theorem 2.1 simplifiesthese to MTopSpin ∧ ( B Z / σ − and MTopSpin ∧ ( B Z / σ − , respectively. Lemma 4.3.
There are equivalences MTopSpin ∧ ( B Z / σ − ≃ MTTopPin − and MTopSpin ∧ ( B Z / σ − ≃ MTTopPin + .Proof. There are surjective maps d n : Top n ։ {± } given by assigning to a homeomorphism the automor-phism it defines on H n ( R n , R n \ ∼ = Z . These commute with the inclusions Top n ֒ → Top n +1 , and passingto the colimit defines a map d : Top ։ {± } . This is a topological version of assigning an orthogonal ma-trix its determinant, classifying whether it preserves or reverses orientation. Given a principal Top-bundle P → M , let Det( P ) → M be the line bundle P × Top R → M , where Top acts on R through d . The mapsTop n × O → Top n × Top → Top n +1 allow us to make sense of “ P ⊕ n Det( P )” as a principal Top-bundle.We abuse notation for a moment to say that a G -structure on a principal Top-bundle P → M is areduction of structure group of P from Top to G . Then, just as in the smooth case, there is a naturalequivalence between the set of TopPin − -structures on P and the set of TopSpin structures on P ⊕ Det( P ),and similarly between the set of TopPin + -structures on P and the set of TopSpin structures on P ⊕ P ).The proof is the same as in the smooth case. These equivalences are the only facts we need to know aboutPin ± in order to prove Theorem 3.3 in the smooth setting, so the argument in the topological setting canproceed in the same way. (cid:3) Therefore by Theorem 2.1, our two normal 1-types are equivalent to
MTTopPin ± . The caveat aboutswitching between pin + and pin − when one passes between the tangent and normal bundles still applieshere. Theorem 4.4 (Kirby-Taylor [KT90b, Theorem 9.2]) . (1) Ω TopPin − ∼ = Z / , generated by E . (2) Ω TopPin + ∼ = Z / ⊕ Z / , with RP generating the Z / summand and E generating the Z / summand. ARUN DEBRAY (3)
The map Ω Pin + → Ω TopPin + is identified with a map Z / → Z / ⊕ Z / which surjects onto the firstfactor and does not hit E . (4) The homomorphism S : Ω TopPin + → Ω TopPin − ∼ = Ω Pin − ∼ = Z / sends RP to a generator. Since Z / − case, detectedby the triangulation obstruction. For the pin + case, the same line of reasoning in the proof of Theorem 3.1allows us to reduce to the case when ξ is a topological pin + structure, so we can compute the action ofAut( ξ ) on the generators. Since E is simply connected, it admits a unique topological pin + structure, so isfixed by Aut( ξ ). For RP , every topological pin + structure on arises from a smooth pin + structure, so wecan reuse our work from Theorem 3.1 to conclude the Aut( ξ )-orbit of RP is again ± [ RP ]. Therefore the setof stable diffeomorphism classes is (( Z / / ( x ∼ − x )) × Z /
2, which has ten elements, and the triangulationobstruction and S ′ are together a complete invariant. (cid:3) The totally non-spin case.
By Lemma 4.1, there is only one normal 1-type to worry about.
Theorem 4.5.
Let G be a finite group of order . There are eight stable homeomorphism classes ofunorientable, totally non-spin topological -manifolds with π ( M ) ∼ = G . The triangulation obstruction andthe Stiefel-Whitney numbers w and w are together a complete stable homeomorphism invariant. Again, this can be extracted from a theorem of Davis [Dav05, Theorem 2.3], who uses a different butequivalent set of invariants.
Proof.
Following the same line of reasoning as in Theorem 3.5, Lemma 4.1 tells us we only have one normal1-type, and its Thom spectrum is
MSTop ∧ ( B Z / σ − . Lemma 4.6.
There is an equivalence MTop ≃ MSTop ∧ ( B Z / σ − .Proof. The proof goes through as in the smooth case, since we have a determinant map and the fact thatfor any Top-bundle P → M , P ⊕ Det( P ) is canonically oriented, analogously to the smooth case. (cid:3) So we need to calculate Ω
Top4 . Proposition 4.7. Ω Top4 ∼ = ( Z / ⊕ , with a basis given by the classes of RP , RP × RP , and E . The Stiefel-Whitney numbers w and w and the triangulation obstruction are linearly independent on this bordism group.Proof. Draw the Atiyah-Hirzebruch spectral sequence computing Ω
Top4 as Ω
STop4 (( B Z / σ − ). It collapsesfor degree reasons in total degree 4 and below, and the 4-line of the E ∞ -page has order 8. Therefore itsuffices to find three linearly independent nonzero elements of Ω Top4 , which can be done by computing w , w , and the triangulation obstruction on RP , RP × RP , and E . (cid:3) Just as in the smooth case, Aut( ξ ) acts trivially. (cid:3) Remark 3.8 also applies in the topological case: the three normal 1-types for unorientable topologicalmanifolds with π ( M ) of order 2 mod 4 are precisely the cases where M has a topological pin + structure, M has a topological pin − structure, and M has neither. References [ABP69] D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson. Pin cobordism and related topics.
Comment. Math. Helv. ,44:462–468, 1969. 6[BG97] Boris Botvinnik and Peter Gilkey. The Gromov-Lawson-Rosenberg conjecture: the twisted case.
Houston J. Math. ,23(1):143–160, 1997. 2[CS71] Sylvain E. Cappell and Julius L. Shaneson. On four dimensional surgery and applications.
Comment. Math. Helv. ,46:500–528, 1971. 2, 6[CS76] Sylvain E. Cappell and Julius L. Shaneson. Some new four-manifolds.
Ann. of Math. (2) , 104(1):61–72, 1976. 2, 6[CS11] Diarmuid Crowley and Jörg Sixt. Stably diffeomorphic manifolds and l q +1 ( Z [ π ]). Forum Math. , 23(3):483–538,2011. https://arxiv.org/abs/0808.2008 . 1[Dav05] James F. Davis. The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds. In
Geometry and topol-ogy of manifolds , volume 47 of
Fields Inst. Commun. , pages 63–76. Amer. Math. Soc., Providence, RI, 2005. 1, 2,6, 8[Deb21] Arun Debray. Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle.2021. https://arxiv.org/abs/2102.02941 . 2
TABLE DIFFEOMORPHISM CLASSIFICATION OF SOME UNORIENTABLE 4-MANIFOLDS 9 [FNOP19] Stefan Friedl, Matthias Nagel, Patrick Orson, and Mark Powell. A survey of the foundations of four-manifold theoryin the topological category. 2019. https://arxiv.org/abs/1910.07372 . 1[Fre82] Michael Hartley Freedman. The topology of four-dimensional manifolds.
J. Differential Geometry , 17(3):357–453,1982. 7[Gia73] V. Giambalvo. Pin and Pin ′ cobordism. Proc. Amer. Math. Soc. , 39:395–401, 1973. 6[GOP +
20] Meng Guo, Kantaro Ohmori, Pavel Putrov, Zheyan Wan, and Juven Wang. Fermionic finite-group gauge theo-ries and interacting symmetric/crystalline orders via cobordisms.
Comm. Math. Phys. , 376(2):1073–1154, 2020. https://arxiv.org/abs/1812.11959 . 2[Gra80] Brayton Gray. Products in the Atiyah-Hirzebruch spectral sequence and the calculation of M SO ∗ . Trans. Amer.Math. Soc. , 260(2):475–483, 1980. 6[HH19] Ian Hambleton and Alyson Hildum. Topological 4-manifolds with right-angled Artin fundamental groups.
J. Topol.Anal. , 11(4):777–821, 2019. https://arxiv.org/abs/1411.5662 . 1[HKT09] Ian Hambleton, Matthias Kreck, and Peter Teichner. Topological 4-manifolds with geometrically two-dimensionalfundamental groups.
J. Topol. Anal. , 1(2):123–151, 2009. https://arxiv.org/abs/0802.0995 . 1[KLPT17] Daniel Kasprowski, Markus Land, Mark Powell, and Peter Teichner. Stable classification of 4-manifolds with 3-manifold fundamental groups.
J. Topol. , 10(3):827–881, 2017. https://arxiv.org/abs/1511.01172 . 1, 7[KPMT20] Justin Kaidi, Julio Parra-Martinez, and Yuji Tachikawa. Topological superconductors on super-string worldsheets.
SciPost Phys. , 9:10, 2020. With a mathematical appendix by Arun Debray. . 2[KPT20] Daniel Kasprowski, Mark Powell, and Peter Teichner. Algebraic criteria for stable diffeomorphism of spin 4-manifolds.2020. https://arxiv.org/abs/2006.06127 . 1[Kre84] M. Kreck. Some closed 4-manifolds with exotic differentiable structure. In
Algebraic topology, Aarhus 1982 (Aarhus,1982) , volume 1051 of
Lecture Notes in Math. , pages 246–262. Springer, Berlin, 1984. 1[Kre99] Matthias Kreck. Surgery and duality.
Ann. of Math. (2) , 149(3):707–754, 1999. https://arxiv.org/abs/math/9905211 . 1, 2, 3[KT90a] R. C. Kirby and L. R. Taylor. A calculation of Pin + bordism groups. Commentarii Mathematici Helvetici , 65(1):434–447, Dec 1990. 5, 6[KT90b] R. C. Kirby and L. R. Taylor.
Pin structures on low-dimensional manifolds , volume 151 of
London Math. Soc.Lecture Note Ser. , pages 177–242. Cambridge Univ. Press, 1990. 2, 6, 7[Kur01] Ichiji Kurazono. Cobordism group with local coefficients and its application to 4-manifolds.
Hiroshima Math. J. ,31(2):263–289, 2001. 1[Ped17] Riccardo Pedrotti. Stable classification of certain families of four-manifolds. Master’s thesis, Max Planck Institutefor Mathematics, 2017. 1[Pet68] F. P. Peterson.
Lectures on Cobordism Theory . Lectures in Mathematics. Kinokuniya Book Store Co., Ltd., 1968. 5[Pol13] Wojciech Politarczyk. 4-manifolds, surgery on loops and geometric realization of Tietze transformations. 2013. https://arxiv.org/abs/1303.6502 . 1[Reu20] David Reutter. Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure. 2020. https://arxiv.org/abs/2001.02288 . 1[Sch76] Martin G. Scharlemann. Transversality theories at dimension four.
Invent. Math. , 33(1):1–14, 1976. 7[Spa03] Fulvia Spaggiari. On the stable classification of Spin four-manifolds.
Osaka J. Math. , 40(4):835–843, 2003. 1[Sto88] Stephan Stolz. Exotic structures on 4-manifolds detected by spectral invariants.
Invent. Math. , 94(1):147–162, 1988.2, 5, 6[Tei92] Peter Teichner.
Topological 4-manifolds with finite fundamental group . PhD thesis, University of Mainz, 1992. https://math.berkeley.edu/~teichner/Papers/phd.pdf . 1[Tei93] Peter Teichner. On the signature of four-manifolds with universal covering spin.
Math. Ann. , 295(4):745–759, 1993.7[Tho54] René Thom.
Quelques propriétés globales des variétés differentiables . PhD thesis, University of Paris, 1954. 6[Wal64] C. T. C. Wall. On simply-connected 4-manifolds.
J. London Math. Soc. , 39:141–149, 1964. 1[Wan95] Zhenghan Wang. Classification of closed nonorientable 4-manifolds with infinite cyclic fundamental group.
Math.Res. Lett. , 2(3):339–344, 1995. 1[WWZ20] Zheyan Wan, Juven Wang, and Yunqin Zheng. Higher anomalies, higher symmetries, and cobordisms ii: Lorentzsymmetry extension and enriched bosonic/fermionic quantum gauge theory.
Ann. Math. Sci. Appl. , 5(2):171–257,2020. https://arxiv.org/abs/1912.13504 . 2
Department of Mathematics, University of Texas, Austin, Texas 78712
Email address ::