Spectral sequences in smooth generalized cohomology
aa r X i v : . [ m a t h . A T ] A ug Spectral sequences in smooth generalized cohomology D ANIEL G RADY H ISHAM S ATI
We consider spectral sequences in smooth generalized cohomology theories, in cluding differential generalized cohomology theories. The main differential spec tral sequences will be of the Atiyah Hirzebruch (AHSS) type, where we provide afiltration by the ˇCech resolution of smooth manifolds. This allows for systematicstudy of torsion in differential cohomology. We apply this in detail to smoothDeligne cohomology, differential topological complex K theory, and to a smoothextension of integral Morava K theory that we introduce. In each case we ex plicitly identify the differentials in the corresponding spectral sequences, whichexhibit an interesting and systematic interplay between (refinement of) classicalcohomology operations, operations involving differential forms, and operations oncohomology with U (1) coefficients.55S25; 55N20, 19L50 Contents
Applications to differential cohomology theories 37 K theory . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Differential Morava K theory . . . . . . . . . . . . . . . . . . . . . . 51 Bibliography 57
Spectral sequences are very useful algebraic tools that often allow for efficient com putations that would otherwise require brute force (see [Mc01] for a broad survey).The Atiyah Hirzebruch spectral sequence (henceforth AHSS) for K theory and anygeneralized cohomology theory, in the topological sense, was introduced by Atiyahand Hirzebruch in [AH62a]. Excellent introduction to the generalized cohomologyAHSS can also be found in Hilton [Hi71] and Adams [Ad74] (Sec. III.7). Other usefulreferences on the subject include Switzer [Sw75] (Sec. 15, from a homology point ofview, including the Gysin Sequence from AHSS) and interesting remarks in relation tospectra are given in Rudyak [Ru08] (Theorem 3.45 (homology), Remark 4.24 (Sheavesand Cech), Remark 4.34 (Postnikov), and Corollary 7.12). A description with an eyefor applications is given in [HJJS08] (Ch. 21).The goal of this paper is to systematically study the spectral sequence in the context ofsmooth or differential cohomology (see [CS85] [Fr00] [HS05] [SS08] [Bu12] [BS09][Sc13]). Existence and interesting aspects of the AHSS in twisted forms of suchdifferential cohomology theories have been considered briefly by Bunke and Nikolaus[BN14], where the main interest was the effect of the geometric part of the twiston the spectral sequence. In this paper we take a step back and consider untwisteddifferential generalized cohomology to systematically study the corresponding AHSSin generality and determine the differentials explicitly as cohomology operations. Fromthe geometric point of view, one might expect on general grounds that the geometricinformation carried by the differential cohomology theory should somehow manifestitself within the spectral sequence. On the other hand, from an algebraic point of view,one might a priori not expect much of that information to be retained, or expect it toeven be totally stripped out while running through the homological algebra machine.We will show that the answer lies somewhat in between, and both intuitions are tosome extent correct: The differentials in the spectral sequence will be essentially2efinements of classical ones, but with additional operations on differential forms.We recently characterized such operations in [GS16a], and so this paper is a naturalcontinuation of that work.Just as generalized cohomology theories are represented by spectra, differential coho mology theories are represented by certain sheaves of smooth spectra called differentialfunction spectra . The original definition of differential function spectra was due to Hop kins and Singer in [HS05], generalized by Bunke, Nikolaus and V ¨olkl in [BNV16] andwas reformulated in terms of cohesion by Schreiber in [Sc13]. The terms smooth co homology and differential cohomology seem to be used interchangeably in some of theliterature (see e.g. [BS10]). However, we will find it useful for us to provide a specificand precise usage, where the first is viewed as being more general than the second. Wealso present most of our ∞ categories as combinatorial, simplicial model categories,rather than quasi categories. We believe that this way nice objects are more easilyand explicitly identifiable, which is something desirable when dealing with differentialcohomology. Indeed, our discussion will be very explicit and the results will be readilyutilizable.Ordinary cohomology has smooth extension with various different realizations, includ ing those of [CS85] [Ga97] [Br93] [DL05] [HS05] [BKS10]. All these realizationsare in fact isomorphic [SS08] [BS10]. A description of K theory with coefficients thatcombines vector bundles, connections, and differential forms into a topological contextwas initiated in [Ka87]. Using Karoubi’s description Lott introduced R / Z valued K theory [Lo94] as well as differential flat K theory [Lo00]. Currently there are variousgeometric models of differential K theory [Lo94] [BS09] [SS08] [FL10] [TWZ13][TWZ15]. As in the case of ordinary differential cohomology these models should beequivalent. Indeed, explicit isomorphisms between various models have been demon strated, for instance between the differential K theory group of [HS05] and [FL10] in[Kl08], between Lott’s R / Z K theory and Lott Freed differential K theory in the latter[FL10], the relation between Bunke Schick differential K theory and Lott( Freed) dif ferential K theory is given in [Ho14], and the isomorphism between Simons Sullivan[SS08] and Freed Lott [FL10] is given in [Ho12].The group structure of differential K theory splits into odd and even degree parts, thusthe refinement preserves the grading. However, the odd part turns out to be moredelicate than the even part. In particular, while any two differential extensions of evenK theory are isomorphic by the uniqueness results of [BS10], odd K theory requiresextra data in order to obtain uniqueness. There are various concrete models in the oddcase, using smooth maps to the unitary group [TWZ13], via loop bundles [HMSV15],and via Hilbert bundles [GL15]. Our results in both even and odd K theory will, of3ourse, not depend on the particular model chosen.Suppose E is a spectrum and X is a space of the homotopy type of a CW complex.Then there is a half plane spectral sequence (AHSS) E p , q ∼ = H p ( X ; E q ( ∗ )) , converging conditionally to E ∗ ( X ). An immediate matter that we encounter in settingup the spectral sequence which calculates the generalized differential cohomology ofa smooth manifold X is how to deal with filtrations. Classically, Maunder [Ma63]gave two approaches to any generalized cohomology theory. The first is by filteringover the q skeletons X q of the topological space X , and second by filtering over thePostnikov systems of spaces Y q , which are the layers of an Ω spectrum associatedto the cohomology theory. Maunder also gives an isomorphism between the twoapproaches. While we expect this to be the case in the differential setting, the proofmight require considerable work. Hence, we leave this as an open problem. Maundersets up his construction in the simplicial complex setting, which is equivalent to settingup in the CW complex setting as the geometric realization of a simplicial set is a CW complex. Simplicial and ˇCech spectral sequences are discussed by May and Sigurdssonin [MS06] (Ch 22).We will prefer the filtration of the spaces/manifolds rather than of the correspondingspectra, as this will naturally bring out the geometry desired in the smooth setting.We first would like to replace a topological space with skeletal filtration by a smoothmanifold and then view this manifold as stack. Hence, in doing this, we need an analogof a skeleton in stacks. This will be done via ˇCech resolution of smooth spaces, andthe replacement of skeletons of a space X will be the various intersections of open setscovering the smooth manifold X .We will use diff( Σ n E , ch) to denote the differential refinement in degree n of a coho mology theory E . This was the notation used in [HS05] and carries more data thanother notation, such as E ( n ). It also avoids possible confusion with other notations,e.g. when dealing with Morava K theory K ( n ) at chromatic level n . The axiomaticapproach is very useful for characterizing a smooth cohomology theory, but one stillneeds the model of [HS05] for actually constructing examples of such smooth spectra.We will be using features of two main approaches at once, namely from [HS05] with I : diff( Σ n E , ch) → E and from [BNV16] [Sc13] with I : E → Π E . Note that E isnot discrete while Π E is, but both are equivalent as smooth spectra E ≃ Π E . Thisessentially boils down to the fact that since Π E is locally constant, the underlyingtheory satisfies Π E ∗ ( U ) = Π E ∗ ( ∗ ) on contractible open sets. On the other hand, thehomotopy invariance of the theory E implies the same thing: namely, E ( U ) ≃ E ( ∗ ),4or a contractible U . These relationships are discussed in further detail in [BNV16].We will be interested in how the differentials look like in our spectral sequences.One might a priori suspect that the differentials in the refined theories should at leastloosely be connected to the differentials of the underlying topological theory. Wewill make this precise below, and so it seems appropriate to understand the formand structure of the differentials in the topological case. To illustrate the point, wewill focus on what might perhaps be the most prominent example, namely the firstdifferential d : H ∗ ( X , K ( ∗ )) → H ∗ ( X , K ( ∗ )) in complex topological K theory K ( X )of a topological space X . This is given by Sq Z [AH62a] [AH62b]. There are exactlytwo stable cohomology operations H ∗ ( X ; Z ) → H ∗ + ( X ; Z ), since H n + ( K ( Z , n )) = Z / n sufficiently large. One of these is zero and the other is β ◦ Sq ◦ ρ , where β is the Bockstein associated to the sequence Z × −→ Z ρ −→ Z with ρ denotingboth the mod 2 reduction and its effect on cohomology with these as coefficients, i.e. ρ : H i ( X ; Z ) → H i ( X ; Z / x ∈ H n ( X ; Z / p ), and with β p the Bocksteinassociated with the sequence Z p × p → Z p ρ p −→ Z p , the elements β p ( x ) is an integralclass in H n + ( X ; Z / p ), i.e. it belongs to the image of the reduction homomorphism ρ p : H n + ( X ; Z ) → H n + ( X ; Z / p ). This can be used to prove the integrality of theclass d ∈ H ( K ( Z / p , Z / p ) as follows (see [FFG86]). The cohomology Serrespectral sequence for the path loop fibration Ω K ( Z , → PK ( Z , → K ( Z ,
3) givesthat H ∗ ( K ( Z , Z / p ) has a single additive generator d in dimension ≤ p . Now wehave a map β : K ( Z / p , → K ( Z ,
3) such that β ∗ ( d ) = d ∈ H ( K ( Z / p , Z / p ),constructed via the Serre spectral sequence of the path loop fibration K ( Z / p , → PK ( Z / p , → K ( Z / p , β induces a map of loop spaces which are alsoSerre fibrations K ( Z / p , / / PK ( Z / p , (cid:15) (cid:15) PK ( Z , (cid:15) (cid:15) K ( Z / p , o o K ( Z / p , / / K ( Z , . The induced homomorphism on the special sequences sends d to d by the constructionof β . Now we have H ( K ( Z / p , Z / p ) = Z / p hence d is contained is containedin the image of the homomorphism ρ p : H ( K ( Z / p , Z ) → H ( K ( Z / p , Z / p ).Therefore d is an integral class. This is attractive as it makes it readily amenable todifferential refinement. 5uch statements, and generalizations to other primes and to other generalized co homology theories, can be made at the level of spectra (see e.g. [Sch]). The firstnontrivial k invariant of connective complex K theory spectrum ku is a morphism k ( ku ) ∈ H ( H Z , Z ), which is equal to β ◦ Sq , where β : H Z / → Σ ( H Z ) isthe Bockstein operator associated to the extension Z × −→ Z −→ Z /
2, and Sq Z isthe pullback of the Steenrod operation Sq ∈ H ( H Z / , Z /
2) along the projectionmorphism ρ : H Z → H Z / ku is a symmetricring spectrum then, by [Sch] (Prop. 8.8), the k invariants are derivations. The onlyderivations (up to units) in the mod p Steenrod algebra A p are the Milnor primitives Q n ∈ H p n − ( H Z / p , Z / p ). At the lowest level we have Q = β p the mod p Bock stein, and the others are realized as k invariants of symmetric spectra, the connectiveMorava K theory spectra k ( n ). That is we have Q n = k p n − ( k ( n )). We will considerrefinements of integral lifts of these.The classical AHSS collapses already at the first page if the generalized cohomologytheory is rational. In fact, it can be shown that for any reasonably behaved spectrumlike all the ones we consider, all the differentials in the AHSS are torsion, i.e. arezero when rationalized (see [Ru08] Cor. 7.12). The differentials in the AHSS in thetopological case are analyzed by systematically by Arlettaz [Ar92]. Using the structureof the integral homology of the Eilenberg MacLane spectra, it is proved there that forany connected space X there are integers R r such that R r d s , tr for all r ≥ s , t . Someaspects of this general feature will continue to hold in the differential setting. Forma homotopy point of view there is not much difference between the localizations at R and at Q . However, from a geometric point of view there is a considerable difference.Nevertheless, we will still use the term “rationalize" when we discuss localization at R , as customary in the homotopy theory literature. We stress that the distinction isneeded in certain geometric settings (see [GM81]), but it will not be an issue for us inthis paper.Note that although the differential cohomology diamond, i.e. the diagram that charac terizes such theories (see Remark 6), certainly detects torsion classes in the flat part ofthe theory, it does not distinguish between torsion at various primes. As a by product,our analysis can be seen as a systematic method for addressing p primary torsion indifferential theories. In [GS16a] we found that the Deligne Beilinson squaring op eration admits lower degree operations refining the Steenrod squares. We have thefamiliar pattern DD , b Sq , b Sq , b Sq , · · · DD , · · · , where DD is the Dixmier Douady class: a non torsion differential cohomology oper 6tion. The refined squares b Sq k + , as the classical squares Sq k , are operations thatare 2 torsion. In this paper we get b Sq k + as we expect, but also differentials d m atlowest degree for every m ,(1–1) d m : Y k Ω k ( M ) −→ H m ( M ; U (1)) . We consider this as a cohomology operation, which can be viewed as first projectingon to the homogeneous component ch m of the Chern character. A U (1) valued ˇCech cocycle is obtained by restricting to 2 m fold intersections of an open cover, pairingwith an appropriate simplex of degree 2 m and exponentiating (This will be spelled outin detail in section 4). If indeed the form ch m arises as the curvature of a bundle, itmust represent a closed form with integral periods. The differential d m can thereforebe understood as the obstruction to this condition. Similar results hold for the odd part,i.e. for differentially refined K theory, where the refined Steenrod square takes thesame form as in differential K theory , while the differentials arising from forms –the analogues of those in (1–1) – are now of odd degrees.The paper is organized as follows. In Sec. 2 we start by carefully setting up thebackground in smooth and differential cohomology, preparing the scene for our con structions. In particular, in Sec. 2.1 we adapt abstract general results on stacks (orsimplicial sheaves) to our context and spell out specific definitions and constructionsthat will be useful for us in later sections; more general and comprehensive accounts canbe found in [Ja87] [Lu11] [Sc13]. Then in Sec. 2.2 we take the approach to differentialcohomology that allows for a direct generalization. Our main constructions will be inSec. 3, and in particular in Sec. 3.1 we provide the filtration via ˇCech resolutions andthen construct the AHSS for smooth spectra in Sec. 3.2 and compare to the AHSS ofthe underlying topological theory. This refinement will depend on whether the degreeis positive, negative, or zero. Then we explore the compatibility of the differentialswith the product structure in Sec. 3.3.Having given the main construction, our main applications of the general spectralsequence to various differential cohomology theories will be presented in Sec. 4. Theconstruction is general enough to apply to any structured cohomology theory whosecoefficients are known. We will explicitly emphasize three main examples: ordinarydifferential cohomology, differential K theory, and a differential version of integralMorava K theory that we introduce. As a test of our method, in Sec. 4.1 we recover theusual hypercohomology spectral sequence for the Deligne complex (see [Br93], [EV92]Appendix), and we do so for manifolds, then products of these, and then more generallyfor smooth fiber bundles. Then the AHSS for K theory is generalized in Sec. 4.2 to7ifferential K theory, where the differential involve refinements of Steenrod squares,in the sense of [GS16a], as well as operations on forms, as indicated above aroundexpression (1–1). We also show that the odd case, i.e. smooth extension of K , leads toa similar construction, but with the differentials now involving odd forms. Then in Sec.4.3 we first introduce a refinement of the integral form of Morava K theory, discussedin [KS03] [Sa10] [SW15], and then characterize the corresponding differentials, whichturn out to have a similar pattern as in K theory, where the operation that gets refined isthe Milnor primitive Q n encountered above. We end with an application to an examplefrom M theory and string theory. Notation.
We have the following morphism that we will use repeatedly throughout.Denote by ρ p : Z → Z / p the mod p reduction on coefficients with correspondingmorphism with the same notation on the cohomology groups with these as coefficients.We will denote by β , β p , and ˜ β the Bockstein homomorphisms associated with thecoefficient sequences 0 → Z → R exp −→ U (1) → , → Z / p × p −→ Z / p ρ p −→ Z / p → , → Z × p −→ Z ρ p −→ Z / p → , respectively. We will let Γ : Z / ֒ → U (1) denote the representation as the square rootsof unity, also with Γ : H n ( − ; Z / → H n ( − ; U (1)) the induced map on cohomology.We will also use more refined Bockstein homomorphisms associated with spectra, andthese will be defined as we need them. In this section we adapt abstract general results on stacks (or simplicial sheaves) to ourcontext and spell out specific definitions and constructions that will be useful for us inlater sections. The interested reader can find more general and comprehensive accountsin [Ja87] [Lu11] [Sc13]. For the reader who is more interested in the applications todifferential cohomology theories, this section can be skipped. However, we wouldlike to emphasize that although the language used in this section is rather abstract, thegenerality gained from this formalism is far reaching and allows this machinery to beused for a wide variety theories, beyond just differential cohomology theories.8ssentially, the axioms characterizing a smooth cohomology theory are not muchdifferent from the axioms characterizing usual cohomology theories. The big differenceis where the theory takes place. More precisely, we want to consider homotopicalfunctors on the category of pointed smooth stacks Sh ∞ ( C art S p) + with C art S p thecategory of Cartesian spaces, rather than the category of pointed topological spaces T op + . Let A b gr be the category of graded abelian groups. Definition 1 (Smooth cohomology) Let E ∗ : Sh ∞ ( C art S p) op + → A b gr be a functorsatisfying the following axioms:(1) (Invariance) E ∗ sends equivalences toisomorphisms.(2) (Additivity) For small coproducts (i.e. ones forming sets) of pointed stacks, W α X α ,wehave E ∗ (cid:16) _ α X α (cid:17) = Y α E ∗ ( X α ) . (3) (Mayer Vietoris) Foranyhomotopy pushout of pointed stacks, Z / / (cid:15) (cid:15) Y (cid:15) (cid:15) X / / X ∪ Z Y , the induced sequence E ∗ ( X ∪ Z Y ) → E ∗ ( X ) ⊕ E ∗ ( Y ) → E ∗ ( Z )is exact.(4) (Suspension) Foranystack X ,there is anisomorphism E n + ( Σ X ) ≃ E n ( X ).Thenwecall E ∗ a smooth cohomology theory. Remark 1
Note that the Mayer Vietoris axiom implies the usual Mayer Vietorissequence. Indeed, let M be a manifold and let V be a local chart of M . Let U be anopen set such that { U , V } isacover of M . Thenthe strict pushout U ∩ V / / (cid:15) (cid:15) V (cid:15) (cid:15) U / / U ∪ V isactuallyahomotopypushout. Wecanequivalently writethisdiagramasahomotopycoequalizer U ∩ V / / / / U ` V / / U ∪ V ,
9n which the homotopy cofiber of the second map can be identified with Σ U ∩ V . Byiterating this argument and applying E ∗ to the the resulting diagram one obtains thelong exact sequence . . . → E ∗ ( U ∩ V ) → E ∗ ( M ) → E ∗ ( U ) ⊕ E ∗ ( V ) → E ∗ + ( U ∩ V ) → . . . , which isthe familiar Mayer Vietoris sequence.The above axioms can be taken as a generalization of the Eilenberg Steenrod axioms(see [Ad74] [Hi71]), where the Mayer Vietoris axiom subsumes both the excisionaxiom and the long exact sequence axiom. It is interesting to note that the axioms donot require homotopy invariance. Namely, if two manifolds M and N are homotopic ,they may fail to be equivalent as stacks. In fact, an equivalence of stacks requires, inparticular, that for every sheaf F (embedded as a stack), we have an isomorphism F ( N ) ≃ π Map( N , F ) ≃ π Map( M , F ) ≃ F ( M ) . In particular, we can take the sheaf of smooth R valued functions on a manifold. Thenif every homotopy equivalence f : M → N induced an equivalence of stacks, we wouldhave an induced isomorphism f ∗ : C ∞ ( N ; R ) → C ∞ ( M ; R ) . Taking N = ∗ and M = R n immediately gives a contradiction. On the other hand,every equivalence of stacks does produce a weak homotopy equivalence of geometricrealizations. To see this, simply note that the geometric realization functor Π : Sh ∞ ( C art S p) → s S et , being a Quillen functor, has a derived functor by Ken Brown’s Lemma [Br73]. Ittherefore preserves weak equivalences between fibrant objects. But these objects areexactly those that satisfy descent, namely stacks, (e.g. manifolds) [Sc13] [Du01]. Remark 2
Givenasmoothcohomologytheory E ∗ ,wealwaysgetapresheafofgradedabelian groups on the site C art S p by precomposing withthe Yoneda embedding: E ∗ : C art S p (cid:31) (cid:127) Y / / Sh ( C art S p) (cid:31) (cid:127) sk / / Sh ∞ ( C art S p) + / / Sh ∞ ( C art S p) + E ∗ / / A b gr , where sk embeds a sheaf as a discrete simplicial sheaf. We will use this fact later inthe construction ofthe spectral sequence in theorem 11.Just as all cohomology theories are representable by Ω spectra, via Brown repre sentability, all smooth cohomology theories are representable by smooth spectra. This10ollows from the version of Brown representability formulated by Jardine in [Ja87]applied to the stable homotopy category of smooth stacks. We will quickly review thebasic properties of this category (see [Lu11] [Ja15]) to establish where our objects ofinterest live.We first recall some operations on stacks that are counterparts to standard operationson topological spaces. Let X and Y be two pointed stacks.(i) The wedge product X ∨ Y is defined via the pushout diagram Y / / Y ∨ X ∗ O O / / X . O O (ii) The smash product X ∧ Y is defined as the quotient X ∧ Y : = X × Y / X ∨ Y ofthe Cartesian product by the wedge product.(iii) The suspension Σ X is defined via the homotopy pushout diagram X (cid:15) (cid:15) / / ∗ (cid:15) (cid:15) ∗ / / Σ X . (iv) The looping, i.e. loop space, Ω X is defined via the homotopy pullback Ω X (cid:15) (cid:15) / / ∗ (cid:15) (cid:15) ∗ / / X . Definition 2
We define the stabilization Stab( Sh ∞ ( C art S p) + ) of smooth pointedstacks tobe the following category: ◦ Theobjects of Stab( Sh ∞ ( C art S p) + ) are sequences ofpointed stacks { E n } ⊂ Sh ∞ ( C art S p) + , n ∈ Z equipped with maps σ n : Σ E n → E n + . ◦ Themorphismsbetween E and F aredefinedtotobethelevelwisemorphisms E n → F n ,commuting with the σ n ’s.This category carries a stable model structure given by first taking the projective modelstructure on sequences of stacks and then performing Bousfield localization with respectto stable weak equivalences in the usual way. This process is described in detail in11Ja87] [Lu11] [Ja15] and we summarize the relevant results found there. The categoryStab( Sh ∞ ( C art S p) + ) admits a stable, closed, simplicial model structure in which ◦ The weak equivalences are stable weak equivalences. That is, a morphism ofsmooth spectra f : E • → F • is a weak equivalence if and only if it induces aweak equivalence Q ( f ) : lim i →∞ Ω i E n + i → lim j →∞ Ω j F n + j . ◦ The fibrant objects are precisely the smooth Ω Spectra, that is, the sequence ofstacks X • whose structure maps σ n : Σ E n → E n + induce equivalences E n ∼ → Ω E n + . Remark 3
We will refer to the stable model category Stab( Sh ∞ ( C art S p) + ) as thecategory of smooth spectra and denote it by Sh ∞ ( C art S p; S p) : = Stab( Sh ∞ ( C art S p) + ) . Example 1
Let M ∈ Sh ∞ ( C art S p) + be a manifold, viewed a stack and equippedwith a basepoint. We can define the smooth spectrum Σ ∞ M in the usual way, as thesequence of suspensions of the manifold M . Given a smooth Ω spectrum E , we candefine asmooth cohomology theory E ∗ ,by setting E q ( M ) ≃ π Map( Σ − q Σ ∞ M , E ) . Differential cohomology theories are examples of the theories introduced above, al though it may not be immediately apparent where the differential cohomology “di amond" diagram [SS08] fits into this context. In fact, it was observed by Bunke,Nikolaus and V ¨olkl in [BNV16], that the diamond provides a further characterizationof all smooth cohomology theories in terms of refinement of topological theories. Thischaracterization happens in addition to the Brown representability described above,and happens only when the category of stacks exhibits so called cohesion . We nowreview the properties of the cohesive structure on smooth stacks [Sc13] that we need,along with the characterization of smooth cohomology theories described in [BNV16].It is shown in [Sc13] that the category Sh ∞ ( C art S p) admits a quadruple ∞ categoricaladjunction ( Π ⊣ disc ⊣ Γ ⊣ codisc)(2–1) Sh ∞ ( C art S p) Γ / / Π / / s S et codisc o o disc o o Π preserves finite ∞ limits and the functors disc and codisc are fully faithful.One implication of this is that s S et embeds into Sh ∞ ( C art S p) as an ∞ subcategoryin two different ways, one reflective, the other reflective and coreflective. From thereflectors one can produce two monads and one comonad defined as follows: Π : = Π ◦ disc , ♭ : = disc ◦ Γ , ♯ : = codisc ◦ Γ . These monads fit into a triple ∞ adjunction ( Π ⊣ ♭ ⊣ ♯ ) which is called a cohesive adjunction. Remark 4
Each monad in the cohesive adjunction picks out a different part of thenature of a smooth stack. This nature is perhaps best exemplified by how the adjointsbehave on smooth manifolds (viewed as stacks). More precisely, if M is a smoothmanifold then, for instance,(i) the comonad ♭ takes the underlying set of points of the manifold and thenembedds this set back into stacks as a discrete object. This functor thereforemisses the smooth structure of the manifold and treats it instead as a discreteobject.(ii) Themonad Π essentially takesthesingularnerveofthemanifloldusing smoothpaths and higher smooth simplices on the manifold. It therefore retains thegeometry of the manifold and “knows" that the points of the manifold ought tobe connected together inasmooth way.The following observation on lifting from simplicial sets to spectra is known ([Sc13],Prop. 4.1.9), but we supply a proof for completeness. Proposition 3
The ∞ adjunction (2–1) lifts toan ∞ adjunction Sh ∞ ( C art S p; S p) Γ s / / Π s / / S p codisc s o o disc s o o on the stable ∞ category of smooth spectra. Moreover, the adjoints satisfy the samecondition asthe ∞ adjunction (2–1) does. Proof.
The category of smooth stacks is presented by the combinatorial simplicialmodel category Sh ∞ ( C art S p) = [ C art S p , s S et] loc , proj , where loc denotes the Bousfield localized model structure at the maps out of ˇCechnerves. The quadruple adjunction is presented by Quillen adjoints ( Π ⊣ disc ⊣ Γ ⊣ S p) disc and codisc both preservehomotopy limits. Hence for a local weak equivalence f : E → F of spectra, we havelim i →∞ Ω i disc( E ) n + i ≃ disc (cid:0) lim i →∞ Ω i F n + i (cid:1) ≃ disc (cid:0) lim j →∞ Ω j F n + j (cid:1) ≃ lim j →∞ Ω j disc( F ) n + j and disc( f ) induces a weak equivalence Q (disc( f )). Hence, disc( f ) is a weak equiv alence. In the same way, codisc preserves local weak equivalences. It follows bythe basic properties of Bousfield localization that disc and codisc are right Quillenadjoints. Again, by the axioms of a closed model category, it follows that the entireadjunction holds as Quillen adjunction of stable model categories. (cid:3) Remark 5
The proof of the previous proposition implies that both disc and codiscpreserve Ω spectra. However, Π and Γ need not take Ω spectra to Ω spectra. Thisproblemcanberemediedbytaking Π s (or Γ s )tobethecompositeof R ◦ Π (or R ◦ Γ ),where R is the fibrant replacement in spectra. Since R defines a left ∞ adjoint tothe inclusion of fibrant objects (and preserves finite ∞ limits), we will still have anadjunction at the level of ∞ categories (although this is not presented by a Quillenadjunction).As in the case of smooth stacks, the quadruple adjunction in the Proposition 3 producesadjoint monads ( Π s ⊣ ♭ s ⊣ ♯ s ) exhibiting stable cohesion. The main observation in[BNV16], recast in the cohesive setting in [Sc13], is the following. Let j : ♭ s → id bethe counit of the comonad ♭ s , and let I : id → Π s be the unit of the monad Π s . Let14 ∈ Sh ∞ ( C art S p; S p) be a smooth spectrum. Then E sits inside a hexagon diagram(2–2) fib( η )( E ) ' ' PPPPPPPPPPPPPP / / cofib( ǫ )( E ) ' ' PPPPPPPPPPPP Σ − Π s cofib( ǫ )( E ) ♥♥♥♥♥♥♥♥♥♥♥ ( ( PPPPPPPPPPPPP E I ( ( PPPPPPPPPPPPPPP ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Π s cofib( ǫ )( E ) ,♭ s E j ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ / / Π s E ♥♥♥♥♥♥♥♥♥♥♥♥♥ where the diagonals are fiber sequences (by definition), the top and bottom sequencesare fiber sequences, and the two squares in the hexagon are homotopy Cartesian, i.e.both are homotopy pullback squares and hence homotopy pushout (via the equivalenceof the two in the stable setting). The latter property is key, because it is a homotopyCartesian square, as on the right of the hexagon, which Hopkins Singer [HS05] tookas the definition of differential cohomology (for a specific choice of the object ofdifferential forms). Bunke Nikolaus V ¨olkl [BNV16] observed that by the hexagon, every smooth spectrum satisfies this kind of Hopkins Singer definition, if one justallows more general objects of differential forms, which is the object cofib( ǫ )( E ) inour notation above.It often happens in practice that the smooth spectra fib( η )( E ) and cofib( ǫ )( E ) containno information away from degree 0. In particular, it often happens that for n > π n Map (cid:0) M , cofib( ǫ )( E ) (cid:1) ≃ , (2–3) π − n Map (cid:0) M , fib( η )( E ) (cid:1) ≃ . (2–4)In this case the E cohomology of a manifold can be calculated as either the flatcohomology or the underlying topological cohomology in all degrees but 0. This issummarized as the following result. Proposition 4
Let E be a smooth spectrum such that (2–3) and (2–4) are satisfied.Thenthe E theory ofa manifold M isgiven by E n ( M ) : = ( Π s E ) n ( M ) n > , ( ♭ s E ) n ( M ) n < , (where indegree 0, E ( M ) isalready characterized by thediamond (2–2)). Proof.
Since the diagonals of the diamond are fiber sequences, they induce long exact15equences in cohomology. Let n be a positive integer. The sequence ♭ s E → E → cofib( ǫ )( E )gives the section of the long sequence π n + Map( M , cofib( ǫ )( E )) → ♭ s E − n ( M ) → E − n ( M ) → π n Map( M , cofib( ǫ )( E )) . By assumption the leftmost and rightmost groups are 0. We therefore have an isomor phism ( ♭ s E ) − n ( M ) ≃ E − n ( M ) . Similarly, the sequence fib( η )( E ) → E → Π s E , gives the long sequence π − n Map( M fib( η )( E )) → E n ( M ) → ( Π s E ) n ( M ) → π − n − Map( M , fib( η )( E )) , and again we get the desired isomorphism. (cid:3) The main applications we have in mind, as we indicated in the Introduction, concern differential cohomology theories . In this section we review some of the conceptsestablished in [Bu12] [BNV16] [Sc13] (which generalize [SS08]), adapted to ourcontext.
Definition 5
Let E ∗ be a cohomology theory. A differential refinement b E ∗ of E ∗ consists of the following data:(1) Afunctor b E ∗ : Sh ∞ ( C art S p + ) op → A b gr ;(2) Three natural transformations:(a) Integration: I : b E ∗ → E ∗ ;(b) Curvature: R : b E ∗ → Z ∗ ( Ω ∗ ⊗ E ∗ ( ∗ ));(c) Secondary Chern character: a : Ω ∗ ⊗ E ∗ ( ∗ )[1] / im( d ) → b E ∗ ;such that the following axioms hold: 16 (Chern Weil). Wehave acommutative diagram b E ∗ R / / I (cid:15) (cid:15) Z ∗ ( Ω ∗ ⊗ E ∗ ( ∗ )) q (cid:15) (cid:15) E ∗ ch / / H ∗ ( Ω ∗ ⊗ E ∗ ( ∗ )) , where ch isthe Chern character map. ◦ (Secondary Chern Weil). Wehave a commutative diagram Ω ∗ ⊗ E ∗ ( ∗ )[1] / im( d ) d / / a ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ Z ∗ ( Ω ∗ ⊗ E ∗ ( ∗ )) b E ∗ R rrrrrrrrrrrr and an exact sequence . . . → E ∗ [1] → Ω ∗ ⊗ E ∗ ( ∗ )[1] / im( d ) → b E ∗ → E ∗ → . . . . Note that in item
Chern Weil above, H ∗ ( Ω ∗ ⊗ E ∗ ( ∗ )) appears as the codomain of theChern character. As explained in [BNV16], this becomes a locally constant stackequivalent to just the locally constant stack on the rationalization of E ∗ , i.e., ch isequivalent to ch : E ∗ → E ∗ ∧ H R (or M R ). Remark 6
The above characterization can ultimately be summarized by saying thatdifferential cohomology fitsinto anexact diamond Ω ∗ ⊗ E ∗ ( ∗ )[1] / im( d ) a ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ d / / Z ∗ ( Ω ∗ ⊗ E ∗ ( ∗ )) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ E ∗− ⊗ R ❧❧❧❧❧❧❧❧❧❧❧❧❧ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ b E ∗ I ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ E ∗ ⊗ R , E ∗− R / Z ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ β E / / E ∗ ch ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ wherethediagonal,topandbottomsequencesareallpartoflongexactsequences. Thebottomsequence isobtained byobserving thatthecofiberoftherationalization mapisan MU (1) (Eilenberg Moorespectrum),whereweidentify R / Z with U (1) throughout.That is, we have a cofiber sequence involving the unit map from the sphere spectrum S = M Z S → M R → MU (1) . E ,weobtain a “Bockstein sequence" E → E ∧ M R → E ∧ MU (1) β E −→ Σ E . Wedefine the flattheory as E U (1) : = E ∧ MU (1)and the rational theory as E R : = E ∧ M R . Remark 7
Differentialcohomologytheoriesareaspecialcaseofsmoothcohomologytheories,whiledifferential functionspectraareaspecialcaseofsmoothspectra. Thus,this section can be viewed asdescribing aspecial case ofthe previous section.Since differential cohomology theories will arise as certain homotopy pullbacks (inDef. 7 below), we will first need to establish the components of the pullback. We beginwith the following lemma that can be found in [Bu12] (Lemma 6.10), which explainshow we can transition from a topological cohomology theory to a smooth one, in aprocess whose direction is opposite to that of the map I . Lemma 6
Let E be a spectrum and define the smooth presheaf of spectra E via theassignment Objects : U Map( Σ ∞ U , E ) , Morphisms : ( f : U → V ) ( f ∗ : Map( Σ ∞ V , E ) → Map( Σ ∞ U , E )) . Then E satisfies descent. Proof.
Let C • ( { U α } ) denote the ˇCech nerve of a good open cover { U α } of somemanifold M . The Yoneda Lemma and basic properties of the mapping space functorimply that we have the following sequence of equivalences E ( M ) : = Map( Σ ∞ M , E ) ≃ Map( Σ ∞ hocolim ∆ op C • ( { U α } ) , E ) ≃ Map(hocolim ∆ op Σ ∞ C • ( { U α } ) , E ) ≃ holim ∆ op Map( Σ ∞ C • ( { U α } ) , E ) ≃ holim n . . . / / / / / / Q αβγ Map( Σ ∞ U αβγ , E ) o o o o o o o o / / / / Q αβ Map( Σ ∞ U αβ , E ) o o o o o o / / Q α Map( Σ ∞ U α , E ) o o o o o ≃ holim n . . . / / / / / / Q αβγ E ( U αβγ ) o o o o o o o o / / / / Q αβ E ( U αβ ) o o o o o o / / Q α E ( U α ) o o o o o , E satisfies descent. (cid:3) The other components of the pullback we want to establish are presented by sheavesof chain complexes. There is a general functorial construction by which one can turnan unbounded chain complex into a spectrum, which we now describe (See [Sh07] fordetails). This functor is called the
Eilenberg MacLane functor(2–5) H : C h → S pand acts on objects as follows. Let C • be an unbounded chain complex, and let Z n denote the subgroup of cycles in degree n . The functor H takes C • and forms thesequence of truncated bounded chain complexes C • ( • ) = (cid:0) . . . → C n → C n − → . . . C → Z (cid:1) = C • (0) (cid:0) . . . → C n → C n − → . . . C → Z − (cid:1) = C • (1) (cid:0) . . . → C n → C n − → . . . C − → Z − (cid:1) = C • (2)... (cid:0) . . . → C n → C n − → . . . C − k → Z − k (cid:1) = C • ( k )...The reason for the group of cycles appearing in degree 0 comes from using the right adjoint to the inclusion i : C h + → C h (as opposed to the left). The left adjoint simplytruncates the complex in degree 0, while the right adjoint truncates and then takes onlythe cycles in degree 0.Continuing with our discussion, at each level in the sequence, H applies the Dold Kanfunctor DK : C h + → s S et to the bounded chain complex in that degree. This gives asequence of spaces DK ( C • ( • )) = DK ( C • (0)) DK ( C • (1)) DK ( C • (2))... DK ( C • ( k ))...Since DK preserves looping (being a right Quillen adjoint) and equivalences (being a19uillen equivalence of model categories), we get induced equivalences σ k : DK ( C • ( k )) → Ω DK ( C • ( k − , which turns DK ( C • ( • )) into a spectrum. Example 2
Consider the unbounded chain complex Z [0], with Z concentrated indegree 0. Then H ( Z [0]) ≃ H Z where theright hand side denotes the Eilenberg MacLane spectrum. Example 3
Fixamanifold M and consider the deRhamcomplex Ω ∗ : = (cid:0) . . . → → → Ω ( M ) → Ω ( M ) → . . . Ω k ( M ) . . . (cid:1) , wherethenonzerotermsareconcentrated innegativedegrees. Then H takes Ω ∗ tothespectrum H ( Ω ∗ ( M )) = DK (cid:0) . . . → → → Ω ( M ) (cid:1) DK (cid:0) . . . → → → Ω ( M ) → Ω ( M ) . . . (cid:1) DK (cid:0) . . . → → → Ω ( M ) → Ω ( M ) → Ω ( M ) . . . (cid:1) ... DK (cid:0) . . . → → → Ω ( M ) → Ω ( M ) → . . . → Ω k cl ( M ) . . . (cid:1) ...By the basic properties of the Dold Kan functor, the stable homotopy groups of thisspectrum are computed as π sn H ( Ω ∗ ( M )) ≃ lim k →∞ π k + n DK (cid:0) . . . → → → Ω ( M ) → Ω ( M ) → . . . Ω k cl ( M ) (cid:1) ≃ lim k →∞ H k + n (cid:0) . . . → → → Ω ( M ) → Ω ( M ) → . . . Ω k cl ( M ) (cid:1) . For n > n ≤ n thde Rhamgroups H n dR ( M ).Now the functor H in (2–5) prolongs to a functor on prestacks H : [ C art S p , C h] → [ C art S p , S p] . In fact, using the properties of the Dold Kan correspondence, it is fairly straightforwardto show that this functor preserves local weak equivalences [Br73]. We therefore get a20unctor of smooth stacks(2–6) H : Sh ∞ ( C art S p; C h) → Sh ∞ ( C art S p; S p) . Recall that for an Ω spectrum E , we always have a rational equivalence:r : E ∧ M R → H ( π ∗ ( E ) ⊗ R ) , where M R denotes an Eilenberg Moore spectrum. Now, since we are working over thesite of Cartesian spaces, the Poincar´e lemma implies that the inclusion j : R [0] → Ω ∗ induces an equivalence id ⊗ j : π ∗ ( E ) ⊗ R [0] → π ∗ ( E ) ⊗ Ω ∗ , where π ∗ ( E ) = E ( ∗ ) (which follows from suspension). Definition 7
Let E be a spectrum. For an unbounded chain complex C • , let τ ≤ C • denote the truncated complex τ ≤ C • = (cid:0) . . . → → → C → C − → . . . → C − n → . . . (cid:1) . Adifferential function spectrum diff( E , ch) isahomotopy pullbackdiff( E , ch) / / (cid:15) (cid:15) H (cid:0) τ ≤ Ω ∗ ⊗ π ∗ ( E ) (cid:1) (cid:15) (cid:15) E ch / / H ( Ω ∗ ⊗ π ∗ ( E )) , where ch = j ◦ r and j induces an equivalence j : π ∗ ( E ) ⊗ R [0] ≃ −→ π ∗ ( E ) ⊗ Ω ∗ . Remark 8
Inourdefinition,wehavechosenthecomplex Ω ∗ ⊗ π ∗ ( E ) asthedeRhamcomplex modeling our rational theory. In general the differential function spectrumdepends on this choice and on the equivalence j [Bu12]. For the purposes of clarityand utility, we will always choose this model, although other models can be treatedanalogously. Wedo,however,keepthedependenceonthemap ch explicittoemphasizethis fact. Example 4 (Deligne cohomology) Let E = H ( Z [ n ]) ≃ Σ n H Z be the n fold sus pensionoftheEilenberg MacLane spectrum. Inunboundedchaincomplexes,wehaveanatural isomorphism Z [ n ] ⊗ Ω ∗ ≃ Ω ∗ [ n ] , where Z [ n ] is the sheaf of locally constant integer valued functions in degree n , andthecomplexontherighthandsidehasbeenshifted up n units. Thatis Ω n isindegree21, while Ω is in degree n . Since Σ n H Z is in the image of the Eilenberg MacLanefunctor H and H preserves homotopy pullbacks, thehomotopy pullbackdiff( Σ n H Z , ch) / / (cid:15) (cid:15) H ( τ ≤ Ω ∗ [ n ]) (cid:15) (cid:15) Σ n H Z ch / / H ( Ω ∗ [ n ])ispresented by the homotopy pullback of unbounded chain complexes Z [ n ] × h Ω ∗ [ n ] τ ≥ Ω ∗ [ n ] (cid:15) (cid:15) / / τ ≤ Ω ∗ [ n ] (cid:15) (cid:15) Z [ n ] / / Ω ∗ [ n ] . Bystability, wecan identify the homotopy pullback withthe shifted mapping cone Z [ n ] × h Ω ∗ [ n ] τ ≤ Ω ∗ [ n ] ≃ cone (cid:0) Z [ n ] ⊕ τ ≤ Ω ∗ → Ω ∗ [ n ] (cid:1) [ − . The right hand side is precisely the Deligne complex Z ∞D ( n + H ( Z ∞D ( n + ≃ diff( Σ n H Z , ch) . The underlying theory that this spectrum represents is precisely Deligne cohomology.In fact, by the Dold Kan correspondence, we have an isomorphism of graded abeliangroups π hom C h ( N ( C ( { U i } ) , Z ∞D ( n + ≃ π Map( Σ ∞ M , diff( Σ n H Z , ch)) . Here N denotesthenormalizedMoorecomplex(adjointtotheDold Kanfunctor DK )and C ( { U i } ) denotes the ˇCech nerve of some good open cover of X . The right handside is simply the definition of diff( Σ n H Z , ch) ( M ), while the left hand side is theshifted total complex of the ˇCech Deligne double complex. It therefore computes thedegree n Deligne cohomology H n ( M ; Z ∞D ( n + Definition 8
Let E be aspectrum and letch : E → H ( τ ≤ Ω ∗ ⊗ π ∗ ( E )) , be the Chern character map as in Definition 7. The differential E cohomology of a22anifold isthe smooth cohomology theory withdegree n component b E n ( M ) ≃ diff( Σ n E , ch) ( M ) . Since, for each n , diff( Σ n E , ch) is a smooth spectrum it fits into a diamond diagramof the form (2–2), as established in [BNV16][Sch]. In [BNV16], it was shown thatthe form that this diamond takes is precisely the differential cohomology diamondin Remark 6. In particular, Proposition 4 allows us to calculate the diff( Σ n E , ch)cohomology in degrees away from 0 asdiff( Σ n E , ch) q ( M ) = E n + q ( M ) q > , E n − + qU (1) ( M ) q < . In this section, we describe general machinery to construct an Atiyah Hirzebruchspectral sequence (AHSS) from a smooth spectrum E . We also describe how to comparethis spectral sequence to the classical AHSS spectral sequence for the underlying theory Π E , in nice cases. The trick to describing the spectral sequence is to choose the right filtration on a fixedmanifold. In the local (projective) model structure on smooth stacks, a natural choicearises: namely the ˇCech type filtration on good open covers . This is indeed the mostnatural choice, since the maps which are weakly inverted in the local model structureare precisely those arising from taking the ˇCech nerve of a good open cover of amanifold. That is, we have a weak equivalence w : hocolim n . . . / / / / / / / / ` αβγ U αβγ o o o o o o / / / / / / ` αβ U αβ o o o o / / / / ` α U α o o o → X . We now explicitly describe a filtration on C ( { U i } ). Recall that any simplicial diagram J : ∆ op → Sh ∞ ( C art S p) can be filtrated by skeleta. More precisely, let i : ∆ ≤ k ֒ → ∆ denote the embedding of the full subcategory of linearly ordered sets [ r ], such that r ≤ k . Then i induces a restriction between functor categories (the k th truncation) τ ≤ k : [ ∆ op , Sh ∞ ( C art S p)] −→ [ ∆ op ≤ k , Sh ∞ ( C art S p)] .
23y general abstract nonsense (the existence of left and right Kan extensions), there areleft and right adjoints (sk k ⊣ τ ≤ k ⊣ cosk k )[ ∆ op , Sh ∞ ( C art S p)] τ ≤ k / / [ ∆ op ≤ k , Sh ∞ ( C art S p)] cosk k o o sk k o o . Furthermore, by composing adjoints, we have an adjunction (sk k ⊣ cosk k )[ ∆ op , Sh ∞ ( C art S p)] sk k / / [ ∆ op , Sh ∞ ( C art S p)] . cosk k o o The functor sk k freely fills in degenerate simplices above level k , while cosk k probesa simplicial object with simplices only up to level k (the singular k skeleton). Proposition 9
Let Y • be a simplicial object in Sh ∞ ( C art S p). Then we can filter Y • byskeleta sk Y • → sk Y • → . . . sk k Y • → . . . → Y • . Thehomotopy colimit over Y • ispresented by theordinary colimithocolim ∆ op ( Y • ) ≃ colim k →∞ L colim ∆ op (sk k Y • ) , where L colim istheleftderivedfunctorofthecolimit,hencecomputableuponsuitablecofibrant replacement ofthe diagram . Proof.
Since Sh ∞ ( C art S p) is presented by a combinatorial simplicial model category,the homotopy colimit over a filtered diagram is presented by the ordinary colimit andthe canonical map L colim k →∞ L colim ∆ op (sk k Y • ) → colim k →∞ L colim ∆ op (sk k Y • )is an equivalence. Since homotopy colimits commute with homotopy colimits, we alsohave an equivalence L colim k →∞ L colim ∆ op (sk k Y • ) ≃ L colim ∆ op L colim k →∞ (sk k Y • ) . Again, using the fact that the ordinary colimit over a filtered diagram presents thehomotopy colimit, we have an equivalence L colim ∆ op L colim k →∞ (sk k Y • ) → L colim ∆ op colim k →∞ (sk k Y • ) ≃ L colim ∆ op ( Y • ) . We take this particular model of the homotopy colimit in order to ensure that taking thecolimit of the resulting diagram makes sense. The claim will also hold for other presentationsof the homotopy colimit Remark 9
The above proposition says that the homotopy colimit over the simplicialobjectisfilteredbyhomotopycolimitsofitsskeleta. Inparticular,if M isaparacompactmanifold, wecan fix agood open cover on M and form the simplicial object given byits ˇCech nerve C ( { U i } ) : = . . . / / / / / / / / ` αβγ U αβγ o o o o o o / / / / / / ` αβ U αβ o o o o / / / / ` α U α o o . Thehomotopy colimit overthis object is then filtered by its skeleta.Let us see exactly what the skeleta look like in this case. To this end, we recall that in Sh ∞ ( C art S p) the full homotopy colimit is presented by the local homotopy formulahocolim ∆ op C ( { U i } ) = Z n ∈ ∆ a α ...α n U α ...α n ⊙ ∆ [ n ] . The filtration on this object is given by first truncating the ˇCech nerve and then freelyfilling in degenerate simplices. As a consequence, in degree k we can forget about thesimplices of dimension higher than k . The homotopy colimit over this skeleton is thengiven by a strict colimit over the diagram (3–1) ` α ...α k U α ...α k ⊙ ∆ [ k ] . . . / / / / / / / / ` αβγ U αβγ ⊙ ∆ [2] o o o o o o / / / / / / ` αβ U αβ ⊙ ∆ [1] o o o o / / / / ` α U α o o ⊙ ∆ [0] , where the face and degeneracy maps are induced by the face and degeneracy mapsof ∆ [ k ]. Taking k → ∞ , we do indeed reproduce the coend representing the fullhomotopy colimit C ( { U i } ).We would like to eventually use this filtration to define a Mayer Vietoris like spectralsequence for general cohomology theory E . To get to this step, however, we willneed to identify the successive quotients of the filtration. To simplify notation in whatfollows, we will fix a manifold M with ˇCech nerve C ( { U i } ) and we set X k : = hocolim ∆ op (cid:0) sk k C ( { U i } ) (cid:1) . Then the quotient X k / X k − can be identified from the previous discussion by quoti enting out the face maps at level k described in diagram (3–1). Since the tensor of asimplicial set and a stack is given by the product of the stack with the discrete inclusion25f the simplicial set, we can identify the quotient from the pushout of coends R n < k ` α ...α n U α ...α n × disc( ∆ [ n ]) ∂ (cid:15) (cid:15) / / ∗ R m ≤ k ` α ...α m U α ...α m × disc( ∆ [ m ]) , where ∂ denotes the boundary inclusion. At the level of points (or elements), a simplexin R n < k ` α ...α k U α ...α n × disc( ∆ [ n ]) is given by a pair( ρ, σ ) ∈ a α ...α k − U α ...α k − × disc( ∆ [ k − , which is glued to lower simplices via the face and degeneracy relations.Let us identify where the boundary inclusion takes a generic simplex. Then the quotient X k / X k − will be obtained by gluing these simplices together to a single point. Notethat the face and degeneracy relations imply that simplices of the form ( ρ, s j + σ ) aresent by d j to ( d j ρ, σ ). Since simplices in the image of the face maps are precisely thosewhich are collapsed to a point, we see that( d j ρ, σ ) ∼ ∗ for every σ. We therefore see that each term of the coproduct ` α ...α k U α ...α k is joined to anotherby the inclusion into a lower intersection. These lower intersections are then collapsedto a point yielding the wedge product _ α ...α k U α ...α k ⊂ X k / X k − . Similarly, the simplex ( s j + ρ, σ ) is sent to ( ρ, d j σ ) under d j . We therefore identify thediscrete simplicial sphere in the quotientdisc( ∆ [ k ] /∂ ∆ [ k ]) ⊂ X k / X k − . Finally, the relations imposed by the coend imply that a simplex of the form ( s j ρ, σ )is glued to ( ρ, d j σ ). The former are precisely those simplices in the simplicial spherewhile the later are glued to the point. Similarly, ( ρ, s j σ ) is glued to the point. Thus wehave the following. Lemma 10
Wecan identify thequotient with thesmash product X k / X k − ≃ disc( ∆ [ k ] /∂ ∆ [ k ]) ∧ _ α ...α k U α ...α k ≃ Σ k (cid:16) _ α ...α k U α ...α k (cid:17) . emark 10 (The filtration as a natural choice) Anotherwaytothinkofourfiltrationabove is the following. Let us form a ˇCech nerve of a manifold, then contract all thepatchesandintersectionsinthat ˇCechnerveaspoints,suchastojustobtainasimplicialset. Thenthe Borsuk’s nerve theorem (see[Bj95]forasurvey,[Ha02]Corollary 4G.3,or [Pr06] Theorem 3.21) says that this simplicial set is equivalent – weak homotopyequivalent –tothesingular simplicial complex ofthemanifold, hencetoitshomotopytype. Moreover, that singular simplicial complex (or rather its geometric realization),inturn,givesaCW complexrealization oftheoriginalmanifold. Sowiththisinmind,onemayviewourfiltrationaboveasthenaturalsmoothrefinementofthefilterationbyCW stagesofthemanifold. Thatis,intakingthe ˇCechnervewithout contractingallitspatchestopoints,weretainexactlythesmoothinformationthat,viaBorsuk’stheorem,corresponds toeachcellinthecanonical CW complexincarnation ofthemanifold. Soin this sense, our refinement can be viewed as the canonical smooth refinement of thetraditional filtering by CW stages.We are now ready to describe the spectral sequence.
Theorem 11 (AHSS for general smooth spectra) Let M be acompact smooth man ifold and let E be asmooth spectrum. There isaspectral sequence with E p , q = H p ( M , E q ) = ⇒ E p + q ( M ) . Here H p denotesthe p th ˇCechcohomologywithcoefficientsinthepresheaf E q . More over, the differential on the E page is given by the differential in ˇCech cohomology. Proof.
The proof is almost immediate from the definitions. Recall that we haveidentified the quotients in Lemma 10. By the axioms for a smooth cohomology theory,we have that the E cohomology of the quotient is given by E ∗ ( X k / X k − ) ≃ E ∗ (cid:16) Σ k (cid:16) _ α ...α k U α ...α k (cid:17)(cid:17) ≃ E ∗− k (cid:16) _ α ...α k U α ...α k (cid:17) ≃ M α ...α k E ∗− k (cid:16) U α ...α k (cid:17) . Applying E p + q to the cofiber squence X p ֒ → X p + ։ X p + / X p gives the long exactsequence in E cohomology(3–2) . . . E p + q ( X p + / X p ) → E p + q ( X p + ) → E p + q ( X p ) → E p + q + ( X p + / X p ) . . . . E p , q term E p , q = M α ,...,α p E q ( U α ...α p ) . Now we want to show that the differential on this page is given by the ˇCech differential δ : E p , q = M α ...α p E q ( U α ...α p ) −→ M α ...α p + E q ( U α ...α p + ) = E p + , q . To this end, note that differential on the E page, by definition, comes from the exactsequence . . . → E p + q ( X p + / X p ) j → E p + q ( X p + ) i → E p + q ( X p ) ∂ → E p + q + ( X p + / X p ) → . . . . We need to show that ∂ j = d = δ is the ˇCech differential. By naturality of theconnecting homomorphism ∂ , we have a commutative diagramˇ C p − ( M ; E q ) ≃ (cid:15) (cid:15) d / / ˇ C p ( M ; E q ) ≃ (cid:15) (cid:15) L α ...α p − E q ( U α ...α p − ) ≃ (cid:15) (cid:15) / / L α ...α p E q ( U α ...α p ) ≃ (cid:15) (cid:15) E p + q − ( X p − / X p − ) j / / (cid:15) (cid:15) E p + q − ( X p − ) ∂ / / (cid:15) (cid:15) E p + q ( X p / X p − ) (cid:15) (cid:15) E p + q − ( ∂ ∆ [ p ] × U α ...α p − ) id / / E p + q − ( ∂ ∆ [ p ] × U α ...α p − ) ∂ / / E p + q (cid:16) ∆ [ p ] /∂ ∆ [ p ] ∧ U α ...α p (cid:17) , where the vertical bottom maps are induced from the inclusion of a factor(3–3) ∆ [ p ] × U α ...α p (cid:31) (cid:127) / / X p ∂ ∆ [ p ] × U α ...α p − (cid:31) (cid:127) / / O O X p − O O ∅ / / O O X p − O O into the p level of the filtration. Comparing the top and bottom composite morphismsin the big diagram, we see that on ( p −
1) fold intersections U α ...α p − , the map d isforced to map a section to the alternating sum of restrictions, as this is precisely the28ap induced by the boundary inclusion in (3–3).All that remains is the convergence. To establish that, we simply note that compactnessimplies that, for large values of p , we have an equivalence X p ≃ X . Moreover, there areonly finitely many diagonal entries at each page of the sequence. With this assumption,the convergence to the corresponding graded complex E p , q ∞ = ker (cid:0) E p + q ( X ) → E p + q ( X p ) (cid:1) ker (cid:0) E p + q ( X ) → E p + q ( X p + ) (cid:1) = F p E p + q ( X ) F p + E p + q ( X )follows exactly as in the classical case in [AH62a]. (cid:3) Fiber bundles.
We can also construct a spectral sequence for a fiber bundle F → N p → M , where each map is a smooth map of manifolds and M is compact. To that end, we notethat for a fixed good open cover { U i } of M , the pullbacks { p − ( U i ) } define a goodopen cover of N . By local triviality, we have that each p − ( U i ) ≃ F × U i . Then, usingthe filtration X k = hocolim ∆ op (cid:0) sk k C ( { p − ( U i ) } ) (cid:1) on the total space N , we identify the successive quotients X k / X k − ≃ Σ k _ α ...α k U α ...α k ∧ F . A similar argument as in the proof of Theorem 11 gives
Theorem 12 (Smooth AHSS for fiber bundles) Let M , N and F be manifolds, with M compact. Let F → N p → M be a fiber bundle. Let E be a sheaf of spectra. Thenthere is aspectral sequence E p , q = H p ( M , E q ( − ∧ F )) = ⇒ E p + q ( N ) . Here H p denotesthe p th ˇCechcohomologywithcoefficientsinthepresheaf E − q ( − ∧ F ) . Remark 11 (Unreduced theories) Notethatthesmooth spectral sequence worksforreduced theories. Onecan treat unreduced theories similarly by setting E q ( M , ∗ ) : = ˜ E q ( M + ) , M + isthepointed stackwithbasepoint ∗ . Inthiscase,wehavetheslightmodificationonthesecondspectralsequence, whichtakes the form E p , q = H p ( M , E q ( − × F )) = ⇒ E p + q ( N ) . Our next task will be to show that these spectral sequences do indeed refine the classicalAtiyah Hirzebruch spectral sequence (AHSS) [AH62a]. Since any smooth theory E comes as a refinement of the underlying topological theory Π E , we will immediatelyget a morphism of spectral sequences induced by the morphism of spectra I : E → Π E . Unfortunately, this morphism does not allow us to compare the differentials of thespectral sequences in the way that we would ideally hope for. However, as we willprogressively see, the situation can be remedied by constructing a slightly differentmorphism of spectral sequences. This morphism is related to the boundary map ofspectral sequences which occurs when a morphism of spectra induces the 0 map oncorresponding spectral sequences (see [Mi81] for a discussion in the case of the Adamsspectral sequence). We first discuss the morphism induced by I and then constructthis “boundary type" map and prove that it indeed defines a morphism of spectralsequences. Definition 13
Let E p , qn and F p , qn bespectralsequences, thatis,asequenceofbigradedcomplexes E p , qn and F p , qn , n ∈ N . A morphism of spectral sequences isamorphismofbigraded complexes f n : E p , qn → F p , qn , definedforall n > N ,where N issomefixedpositiveinteger. Furthermore,werequirethe map f n + to be the map on homology induced by f n . We call the smallest integer N such that f n are defined for n > N the rank of themorphism.We now apply this to the smooth AHSS. The next result should follow from generalprinciples, but we emphasise it explicitly for clarity and for subsequent use. Proposition 14
Let E and F be smooth spectra. Then a map f : E → F induces amorphism of corresponding smooth AHSS’s E p , qn → F p , qn . roof. Fix a manifold X and a good open cover { U i } . Let X p denote the p th filtrationof the ˇCech nerve as before. It is clear by naturality that a map of spectra f : E → F induces a morphism of long exact sequences (see (3–2)) . . . E p + q ( X p + / X p ) / / (cid:15) (cid:15) E p + q ( X p + ) / / (cid:15) (cid:15) E p + q ( X p ) / / (cid:15) (cid:15) E p + q + ( X p + / X p ) . . . (cid:15) (cid:15) . . . F p + q ( X p + / X p ) / / F p + q ( X p + ) / / F p + q ( X p ) / / F p + q + ( X p + / X p ) . . . . It follows immediately from the construction of the corresponding exact triangles thatthis morphism commutes with the differentials. (cid:3)
This now allows us to compare the topological and the smooth theories.
Corollary 15
Let E be a smooth spectrum and Π E be the underlying topologicaltheory. Let E n and F n denote the spectral sequences corresponding to E and Π E ,respectively. Thenatural map I : E → Π E induces amorphism ofclassical AHSS’s I : E p , qn → F p , qn . Remark 12
It is interesting to note that the smooth spectrum Π E is, by definition,locally constant. From the discussion around (2–1), this means that we have isomor phism Π E q ( U ) ≃ π − q Map( U , Π E ) ≃ π − q Map( ∗ , Π E ) ≃ π − q Π E ≃ Π E q ( ∗ )for every element of a good open cover (or higher intersection) U . This connects,via Borsuk’s theorem mentioned in Remark 10 above, the “smooth AHSS for locallyconstant coefficients" with the classical AHSS: the locally constant coefficients seeeach(contractible) patchasapoint,andhencebyBorsuk’stheoremtheyseeour“ˇCechfilteration" to bethe classical CW cell filteration.From the construction of our smooth AHSS, we immediately get that the spectralsequence associated to the smooth spectrum is a refinement of the classical topologicalAHSS. Corollary 16
The spectral sequence F p , qn is precisely the AHSSfor the cohomologytheory Π E . Here we have an unfortunate conflict of notation. We are using the same symbols for thepages in the spectral sequences for both the classical and the refined theories. We will aim tomake the context explicit whenever a possible ambiguity arises.
31e now would like to apply the above machinery to differential cohomology theories.In particular, we note that for a differential function spectrum diff( E , ch), the naturalmap I : diff( E , ch) → E , which strips the differential theory of the differential data and maps to the bare underly ing theory, is precisely the map induced by the unit I : id → Π . In the above discussion,we observed that this map always induces a morphism of spectral sequences. More over, the target spectral sequence is exactly the AHSS for the underlying topologicaltheory. One might hope to be able to use this map to compare the differentials in therefined theory with those differentials in the classical AHSS.Unfortunately, this does not work in practice, as we will see when we discuss appli cations in Sec. 4. The core issue is that the spectral sequence for the refined theoryusually ends up shifted with respect to the classical AHSS. As a consequence, thenonzero terms in each sequence are interlaced with respect to one another and the map I ends up killing all the nonzero terms. This, in turn, stems from the appearance of theBockstein map (which raises degree by 1) in the differential cohomology diagram.However, there is often a different map between the lower quadrants of the the twospectral sequence corresponding to diff( E , ch) and E , which lowers the degree as tomatch the corresponding nonzero entries. This map is related to the so called boundarymap between spectral sequences studied in [Mi81]. The next proposition concerns thismap and will be essential for comparing the differentials in the refined theory to thoseof the classical theory. Proposition 17 (i)
Let E be a spectrum such that π ∗ ( E ) is concentrated in degreeswhich are a multiple of some integer n ≥ π ∗ ( E ) is projective in those degrees. Then the sequence ofspectra E → E ∧ M R → E ∧ MU (1) β E −→ Σ E , induces ashort exact sequence on coefficients(3–4) 0 → π ∗ ( E ) → π ∗ ( E ) ⊗ R → π ∗ ( E ) ⊗ U (1) → . (ii) Let β denotetheconnectinghomomorphism(i.e. theBockstein)forthecoefficientsequence (3–4). Let E p , qn denotethespectralsequencecorrespondingto Σ − E ∧ MU (1)and let F p , qn denote the spectral sequence corresponding to E . Then β : E p , qn → F p , qn induces amorphism ofspectral sequences ofrank 2.32 roof. Consider the long Bockstein sequence . . . E r −→ E ∧ M R e −→ E ∧ MU (1) β E −→ Σ E . . . , induced by the cofiber sequence S → M R → MU (1) . Fix a manifold M and let X p denote the p level of the ˇCech filtration. Now eachspectrum in the above sequence has a long exact sequence induced be the cofibersequences X p − → X p → X p / X p − . from which one builds the exact couple for for the corresponding spectral sequence.Using the properties of π ∗ ( E ) along with this sequence, we can fit the long exactsequences into a diagramˇ C p ( X ; π − q − ( E )) q ∗ / / r (cid:15) (cid:15) E p + q − ( X p ) i ∗ / / r (cid:15) (cid:15) E p + q − ( X p − ) ∂ / / r (cid:15) (cid:15) (cid:15) (cid:15) ˇ C p ( X ; π − q − ( E R )) q ∗ / / e (cid:15) (cid:15) E p + q − R ( X p ) i ∗ / / e (cid:15) (cid:15) E p + q − R ( X p − ) ∂ / / e (cid:15) (cid:15) (cid:15) (cid:15) ˇ C p ( X ; π − q − ( E U (1) )) q ∗ / / β E (cid:15) (cid:15) E p + qU (1) ( X p ) i ∗ / / β E (cid:15) (cid:15) E p + qU (1) ( X p − ) ∂ / / β E (cid:15) (cid:15) (cid:15) (cid:15) / / E p + q ( X p + ) / / E p + q ( X p ) / / ˇ C p ( X ; π − q + ( E )) , where both the rows and columns are part of exact sequences and ˇ C p ( X ; A ) denotesthe group of ˇCech p cochains with coefficients in A . Since everything commutes, thisinduces a correponding short exact sequence of E pages. At each ( p , q ) entry thissequence is given by0 → C p ( X ; π − q ( E )) → C p ( X ; π − q ( E ) ⊗ R ) → C p ( X ; π − q ( E ) ⊗ U (1)) → . Since the differentials on the E page are precisely the ˇCech differentials, the construc tion of the Bockstein map in ˇCeach cohomology will produce a map of E pages β : H p ( X ; π − q ( E ) ⊗ U (1)) → H p + ( X ; π − q ( E )) . We need to show that this map commutes with the differential. Choose a representative x of a class in H p ( X ; π − q ( E ) ⊗ U (1)). By definition, y = β ( x ) is a class such that33 ( y ) = δ ( x ), where x is such that e ( x ) = x . Then r ( d y ) = d r ( y ) = d δ ( x ) , We want to show that there is a lift z of d x such that δ ( z ) = d δ ( x ). Indeed, if this isthe case, then d y represents β ( d x ) and we are done.To construct z , recall that d x is defined by first pulling back by the quotient q , whichlies in the image of the map induced by the inclusion i : X p ֒ → X p + , and then applyingthe boundary to an element of the preimage. Let w be such that i ∗ ( w ) = q ∗ ( x ) . Chasing the diagramˇ C p ( X ; π − q − ( E R )) q ∗ / / e (cid:15) (cid:15) E p + q − R ( X p ) e (cid:15) (cid:15) e / / E p + q − R ( X p − ) e (cid:15) (cid:15) ˇ C p ( X ; π − q − ( E U (1) )) q ∗ / / β E (cid:15) (cid:15) E p + qU (1) ( X p ) β E (cid:15) (cid:15) i ∗ / / E p + qU (1) ( X p − ) β E (cid:15) (cid:15) / / E p + q ( X p ) i ∗ / / E p + q ( X p − ) , we see that 0 = β E q ∗ ( x ) = β E i ∗ ( w ) = i ∗ ( β E w ). By exactness of the rows, this impliesthat β E w =
0. Therefore, there is a class w ∈ E p + q + R ( X p + ) such that e ( w ) = w .Now, by definition of the differential, we have e ( ∂ w ) = ∂ ( e ( w )) = ∂ w = d x and z : = ∂ w is a lift of d x . Using the fact that δ = d = ∂ q ∗ , we have δ ( z ) = δ ( ∂ w ) = ∂ ( q ∗ ∂ w ) . By exactness, we have i ∗ ( q ∗ ∂ w ) = = q ∗ ∂ q ∗ ( x ) = q ∗ ( δ ( x )) , and it follows from the definition that δ ( z ) = d ( δ ( x )).To show that H ∗ ( β ) commutes with the higher differentials, we proceed by induc tion. The above proves the base case. Suppose β induces a map H n ( β ) on E n whichcommutes with d n . Then H n ( β ) induces a well defined map H n + ( β ) on the E n + page. Let x ∈ T ni = ker( d n + ) be a representative of a class on the E n page. Then bedefinition, H n + ( β )( x ) = β ( x ) and the exact same argument as before (replacing d d n + ), gives the result. (cid:3) Having done the heavy lifting in the above proposition, we will now apply this tostraightforwardly relate the differentials of the refined theory to those of the underlyingtopological theory. This will use an explicit alternative to the map I , along the lines ofthe discussion just before the statement of the above proposition. Theorem 18 (Refinement of differentials) Let E be a spectrum satisfying the prop erties of Proposition 17 and let diff( E , ch) be adifferential function spectrum refining E . Let E n and F n denote the smooth AHSS’s corresponding to diff( E , ch) and E ,respectively. Then the Bockstein β defines a rank 2 morphism of fourth quadrantspectral sequences β : E p , qn → F p , qn , q < . Proof.
Recall that for q <
0, Proposition 4 implies that diff( E , ch) q ( M ) ≃ E q − U (1) ( M ).The claim then follows from the previous proposition. (cid:3) Let E be an E ∞ ring spectrum. Then the associative graded commutative producton E ∗ induces a product (associative and graded commutative) on the refinementdiff( Σ n E , ch) ∗ , that is, a map(3–5) ∪ : diff( Σ n E , ch) k ⊗ diff( Σ m E , ch) j −→ diff( Σ n + m E , ch) k + j (see [Bu12] [Up15]). The goal of this section will be to establish the following veryuseful property, in analogy with the classical case. Proposition 19 (Compatibility with products) Theproduct ∪ : diff( Σ n E , ch) k ⊗ diff( Σ m E , ch) j → diff( Σ n + m E , ch) k + j induces amorphism ofspectral sequences ∪ : E ∗ ( n ) × E ∗ ( m ) → E ∗ ( n + m ) . Moreover, the differentials satisfy the Liebniz rule d ( xy ) = d ( x ) y + ( − p + q xd ( y ) . E page. Recall from theconstruction of the spectral sequence that the E p , q is given by E p , q = M α ...α p diff( Σ n E , ch) q ( U α ...α p ) ≃ ˇ C p ( M ; diff( Σ n E , ch) q ) . Using the product (3–5), we get a cross product map × : M α ...α p diff( Σ n E , ch) q ( U α ...α p ) × M α ...α r diff( Σ m E , ch) t ( U α ...α r ) →→ M α ...α p M α ...α r diff( Σ n + m E , ch) q + t ( U α ...α p × U α ...α r ) . (3–6)We also have an isomorphism M α ...α s diff( Σ n + m E , ch) q + t (( U × U ) α ...α s ) ≃ diff( Σ n + m E , ch) q + t (cid:16) _ α ...α s ( U × U ) α ...α s (cid:17) ≃ diff( Σ n + m E , ch) q + t (cid:16) _ α ...α p _ α ...α r _ p + r = s U α ...α p × U α ...α r (cid:17) ≃ M α ...α p M α ...α r M p + r = s diff( Σ n + m E , ch) q + t ( U α ...α p × U α ...α r ) , given by decomposing the product of the cover { U α } with itself. Finally, we canpullback by the diagonal map ∆ ∗ : M α ...α s diff( Σ n + m E , ch) q + t (( U × U ) α ...α s ) → M α ...α s diff( Σ n + m E , ch) q + t ( U α ...α s ) ≃≃ ˇ C p + r ( M ; diff( Σ n + m E , ch) q + t ) . The cup product on the E page is defined by the composite map ∆ ∗ × . Lemma 20
Thedifferential d onthe E page satisfies the Leibniz rule. Proof.
The construction of the cup product on the E page is precisely the cup productstructure for ˇCech cohomology. The ˇCech differential satisfies the Leibniz rule andthis is precisely d , by construction. (cid:3) We are now ready to prove Proposition 19.36 roof.
The proof follows by induction on the pages of the spectral sequence. The basecase is satisfied by Lemma 20. Now suppose we have a cup product map ∪ : E ( n ) k × E ( n ) k → E ( n + m ) k , such that d k satisfies Leibniz. By definition, we have E ( n ) p , qk + = ker (cid:0) d k : E ( n ) p , qk → E ( n ) p + k , q + k − k (cid:1) im (cid:0) d k : E ( n ) p − k , q − k + → E ( n ) p , q (cid:1) , and we define the cup product ∪ : E ( n ) p , qk + × E ( m ) r , sk + → E ( n + m ) p + r , q + sk + by restricting to elements in the kernel of d k . The product is well defined since d k satisfies the Leibniz rule. At this stage the problem looks formally like the classicalproblem. Hence, analogously to the classical discussion in [Ha02], it is tedious butstraightforward to show that d k + also satisfies the Leibniz rule. (cid:3) In this section we would like to apply the spectral sequence constructed in the previoussection to various differential cohomology theories. The construction is general enoughto apply to any structured cohomology theory whose coefficients are known. We willexplicitly emphasize three main examples. The first two are to known theories, namelyordinary differential cohomology and differential K theory. We take this opportunityto explicitly develop the third theory, which is differential Morava K theory and thenapply our smooth AHSS construction to it.
We begin by recovering the usual hypercohomology spectral sequence for the Delignecomplex (see [Br93], [EV92] Appendix) using our methods. We will first look atmanifolds, then products of these, and then more generally to smooth fiber bundles.Let us consider the smooth spectrum diff( Σ n H Z , ch) representing differential coho mology in degree n . We would like to see what our smooth AHSS gives in this case.We recall that diff( Σ n H Z , ch) is represented by Deligne cohomology of the sheaf ofchain complexes Z ∞D ( n ) via the Eilenberg MacLane functor H : Sh ∞ ( C art S p; C h) → h ∞ ( C art S p; S p) (expressions (2–6)). It follows from the general properties of thisfunctor that the homotopy groups are given by π k diff( Σ n H Z , ch) ≃ H k Z ∞D ( n ) . In this case we have the immediate corollary to Theorem 11.
Corollary 21
Thespectral sequence for Deligne cohomology takes the form E p , q = H p ( X ; H − q Z ∞D ( n )) ⇒ H p + q ( X ; Z ∞D ( n )) , which isessentially the hypercohomology spectral sequence forthe Deligne complex,but shifted as afourth quadrant spectral sequence.For the sake of completeness, we work out this spectral sequence and recover thedifferential cohomology diamond (2–2) from the sequence. This will help to illustratehow the general spectral sequence behaves and how it can be used to calculate generaldifferential cohomology groups.Now over the site of Cartesian spaces, the Poincar´e Lemma implies that we have anisomorphism of presheaves d : Ω n − / im(d) ≃ → Ω n cl . Since Ω n cl is a sheaf over the siteof smooth manifolds, the gluing condition allows us to calculate the relevant terms onthe E page of the spectral sequence:10 Ω n cl ( M )... 0 − ( n −
2) 0 − ( n − H n − ( M ; U (1)) − n d d d H n − ( M ; U (1)) will survive to the E ∞ page and we have an isomorphism H n − ( M ; U (1)) ≃ F n − b H n ( M ; Z ) F n b H n ( M ; Z ) . In fact, it is not hard to see that the definition of the filtration gives F n b H n ( M ; Z ) ≃ H n − ( M ; U (1)) ≃ F n − b H n ( M ; Z ) ֒ → b H n ( M ; Z ) . On the E n page we get one possibly nonzero differential d n : Ω n ( M ) cl → H n ( M ; U (1)) . Proposition 22
The differential d n for the AHSS for Deligne cohomology can beidentified withthe composition Ω n cl ( M ) → H n dR ( M ) R ∆ n −→ H n ( M ; R ) exp −→ H n ( M ; U (1)) , and the kernel isprecisely those formswhich have integral periods. Proof.
We will unpack the definition of the differential in the AHSS in detail. Thisin turn will require unpacking the connecting homomorphism in the Deligne model ofordinary differential cohomology (see [Br93]). Denote by X p the ˇCech filtration, andlet ∂ : diff( Σ n H Z , ch) q ( X p ) → diff( Σ n H Z , ch) q + ( X p + / X p )denote the connecting homomorphism in the long exact sequence associated to thecofiber sequence X p ֒ → X p + ։ X p + / X p in the usual way. In what follows, we willdenote ˇCech Deligne cochains on the p th level of the filtration X p as a p tuple( z , z , . . . , z p ) ∈ b C q ( X p ) , where z i is a ( q − i ) form defined on i fold intersections.Now, by definition, d n : E , n → E n , n is given by d n = ∂ ( j ∗ ) − , where ( j ∗ ) − denotesa choice of element in the preimage of the restriction j ∗ induced by j : X ֒ → X n − .Since we have d k = k < n , the differential d n is defined on all elements z ∈ Ω n cl ( M ). Let g be a locally defined ( n −
1) form trivializing z . Then we canchoose ( j ∗ ) − z to be the ˇCech Deligne cocycle(4–1) ( j ∗ ) − z = ( g , g , g , . . . , g n − ) | {z } n − ∈ b C ( X n − ) , Note that the differential only takes this form at the (0 ,
0) entry. In general, the differentialformed from the n th derived couple will be more complicated g k is a ( n − k −
1) form satisfying the cocycle condition δ ( g k ) = ( − k dg k + . To see where the boundary map takes this element, let y be a ˇCech Deligne cochain given by y = (cid:16) g , g , g , . . . , g n − , exp(2 π ig n − ) (cid:17)(cid:17)| {z } n ∈ b C ( X n ) , where g n − is any smooth R valued function satisfying d ( g n − ) = ( − n − δ ( g n − ).Now y is not ˇCech Deligne closed in general since Dy = ( d + ( − n − δ ) y = (0 , , . . . , exp(( − n − π i · δ ( g n − )))and g n − may not satisfy the cocycle condition δ ( g n − ) =
0. However, by the ˇCech de Rham isomorphism (see for example [BT82]), this element in the ˇCech de Rhamdouble complex is isomorphic to an R valued ˇCech cocycle on n fold intersections.Explicitly, there is a constant R valued cocycle r n such that δ ( g n − ) = r n . It followsfrom the ˇCech singular isomorphism and the singular de Rham isomorphism that theclass of r n can be represented by the singular cocycle given by the pairing R σ z for anycycle σ in M . Since the class R σ z was just an unraveling of the boundary ∂ (( j ∗ ) − z ),we have proved the claim. (cid:3) In the next section, we will need to make use of a differential refinement of theChern character. To this end, we briefly discuss differential cohomology with rationalcoefficients b H n ( − ; Q ). These groups are obtained via the differential function spectradiff( Σ n H Q , ch) which fits into the homotopy cartesian squarediff( H Q , ch) / / (cid:15) (cid:15) H ( τ ≤ Ω ∗ [ n ]) (cid:15) (cid:15) Σ n H Q / / H ( Ω ∗ [ n ]) . As a consequence of Proposition 4, the cohomology groups with values in this spectrumare calculated asdiff( Σ n H Q , ch) q ( M ) = H n + q ( M ) q > , b H n ( M ; Q ) q = H n − + q ( M ; R / Q ) q < . Note that this cocycle condition is necessary for y to be an lift of ( j ∗ ) − z to the n level ofthe filtration Proposition 23
The differential d n on the E n page for the AHSS spectral sequencefor diff( Σ n H Q , ch) is given by Ω n cl ( M ) → H n dR ( M ) R ∆ n −→ H n ( M ; R ) −→ H n ( M ; Q / Z ) , and the kernel isprecisely those formswhich have rational periods.We will make use of this result when we discuss the differentials in smooth K theoryin the next section. For now, from Proposition 22, we immediately get the followingcharacterization of closed forms with integral periods and forms with rational periodsusing our smooth AHSS. Corollary 24 (i)
Thegroup ofclosed formswithintegral periods onamanifold M isgiven by Ω n cl , Z ( M ) ≃ b H n ( M ; Z ) F b H n ( M ; Z ) . (ii) Thegroup ofclosed forms withrational periods on amanifold M isgiven by Ω n cl , Q ( M ) ≃ b H n ( M ; Q ) F b H n ( M ; Q ) . K theory In this section we examine the smooth AHSS for the differential function spectrumdiff( K , ch), corresponding to complex K theory. Proposition 4 allows us to calculatethe cohomology groups on a paracompact manifold M as (see [Lo94] [BS09] [SS08][FL10])(4–2) diff( K , ch) q ( M ) = K q ( M ) , q > , b K ( M ) , q = , K qU (1) ( M ) , q < . The exact argument in the proof of proposition 22 applies, with R / Q in place of R / Z ≃ U (1). K and K U (1) are periodic. Indeed, K U (1) ( M ) fits into an exactsequence . . . → K − ( M ) ⊗ R → K − U (1) ( M ) → K ( M ) → K ( M ) ⊗ R → . . . . Consequently, the periodicity of both integral and rational K theory, along with anapplication of the Five Lemma, imply that K U (1) is 2 periodic. In particular, we have K qU (1) ( ∗ ) ≃ U (1) and K q + U (1) ( ∗ ) ≃ , q ∈ Z . Given the correspondence (4–2), we see that for a contractible open set U , we have anisomorphism diff( K , ch) q + ( U ) ≃ K qU (1) ( ∗ ) ≃ U (1)for q <
0. For degree 0, the differential cohomology diamond in this case takes theform Q k − Ω k − / im( d ) a ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ d / / Q k Ω k cl & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼ K − R qqqqqqqqqq ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ b K I ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ R qqqqqqqqqqqqqq K R . K − U (1) ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ β K / / K ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ This implies that for a contractible open set U , differential K theory b K ( U ) fits intothe short exact sequence0 → Y k − Ω k − / im( d )( U ) → b K ( U ) → Z → . Hence, over the site of Cartesian spaces, we have a naturally split short exact sequenceof presheaves 0 → Y k − Ω k − / im( d ) → b K → Z → . Over that site, the presheaf on the left hand side is actually a sheaf and is naturallyisomorphic (by Poincar´e lemma) to the sheaf Q k Ω k cl . We therefore make the identi fication(4–3) b K ≃ Y k Ω k cl ⊕ Z . Remark 13
Itisimportant tonote thattheidentification (4–3) isonlytrueonthesite42f Cartesian spaces, which is to say that it holds only locally. On the site of smoothmanifolds, this isof course not the case.Next, since both Ω k cl and Z are sheaves on the site of smooth manifolds, we canidentify the degree 0 ˇCech cohomology with these coefficients with the value of thissheaf on M . Isolating the terms on the E page which converge to b K ( M ), we get10 Q k Ω k cl ( M ) ⊕ Z − H ( M ; U (1)) H ( M ; U (1)) − − H ( M ; U (1)) − d We see that all the differentials are zero except for the map labelled d above. On the43 page we get10 ker( d ) − H ( M ; U (1)) H ( M ; U (1)) − − H ( M ; U (1)) d d The higher pages will fall into cases depending on the parity. We observe that for eacheven page E m , there is one non zero differential given by d m . For the odd pages thedifferentials are given by an odd degree U (1) cohomology operation.Note that, in the diagrams, we are interested in the case p + q =
0, correspondingto diagonal entries. Now p ≥
0, as the ˇCech filtrations are of non negative degrees,which implies that q ≤
0. Hence the entries go down the diagonal. Our first goal willbe to identify the even differentials d m . In order to do this, let us recall that thereis a differential Chern character map (see [Bu12] [Sc13]) which is stably given by amorphism of smooth spectra b ch : diff( K , ch) → Y k diff( Σ k H Q , ch) . Post composing this map with the projection pr m onto the 2 m component gives a mapof smooth spectra pr m b ch : diff( K , ch) → diff( Σ m H Q , ch) . Using this map, we can prove the following analogue of Proposition 23.
Proposition 25
The group of permanent cycles in bidegree (0 ,
0) in the AHSS fordiff( K , ch) is a subgroup of even degree closed forms with rational periods. That is,44ehave E , ∞ ⊂ Y k Ω k cl , Q ( M ) ⊕ Z . Proof.
We prove by induction on the even pages of the spectral sequence that, for all n , E , n must be a subgroup of Y k ≤ n Ω k cl , Q ( M ) ⊕ Y k > n Ω k cl ( M ) ⊕ Z . For the base case, observe that the map pr b ch induces a rank 1 morphism of AHSS’sand therefore commutes with d . It is straightforward to check, using the definitions,that this leads to the commutative diagram Q k Ω k cl ( M ) ⊕ Z pr / / d (cid:15) (cid:15) Ω ( M ) d ′ (cid:15) (cid:15) H ( M ; R / Z ) q / / H ( M ; R / Q ) . By Proposition 22, we see that the kernel of d must be a subgroup of Ω , Q ( M ) ⊕ Q k > Ω k ( M ) ⊕ Z .Now suppose the claim is true for d n . Again, we have that pr n + b ch commutes with d n + and we have a commutative diagramker( d n ) pr n + / / d n + (cid:15) (cid:15) Ω n + ( M ) d ′ n + (cid:15) (cid:15) H n + ( M ; R / Z ) q / / H n + ( M ; R / Q ) . By the induction hypothesis,ker( d n ) ⊂ Y k ≤ n Ω k cl , Q ( M ) ⊕ Y k > n Ω k cl ( M ) ⊕ Z , and the kernel of d n + is as claimed. (cid:3) We now turn to the first odd differential d . Recall that β and ˜ β denote the Bocksteinhomomorphisms corresponding to the sequences 0 → Z → R exp −→ U (1) → The differential is 0 for the odd pages, and so no generality is lost by restricting to the evenpages. → Z → Z → Z / →
0, respectively. We still also denote by Γ : H n ( − ; Z / → H n ( − , U (1)) the map induced by the representation of Z / ρ : Z → Z / Proposition 26 (Degree three differential) The first odd degree differential in theAHSSfor differential K theory isgiven by d = b Sq : = Γ Sq ρ β, q < , Sq Z : = ˜ β Sq ρ , q > , , q = . Proof.
The case for q = q >
0, this follows from the fact that theintegration map defines an isomorphism I : diff( K , ch) q ( M ) ≃ −→ K q ( M ) for q > d for the classical AHSS is given by Sq Z and the integrationmap defines an isomorphism of corresponding first quadrant spectral sequences, thecase q > q <
0, Corollary 15 implies that the Bockstein β commutes with the differentialson the E page. We therefore have(4–4) β d = Sq Z β = e β Sq ρ β . Rephrasing, we have the commuting diagram H n − ( M ; U (1)) d / / β (cid:15) (cid:15) H n + − ( M ; U (1)) β (cid:15) (cid:15) H n ( M ; Z ) Sq Z / / H n + ( M ; Z ) . We now claim that ˜ β = β ◦ Γ . Indeed, we have a morphism of short exact sequences Z × / / id (cid:15) (cid:15) Z ρ / / × π i (cid:15) (cid:15) (cid:15) (cid:15) Z / Γ (cid:15) (cid:15) Z × π i / / R exp / / U (1) . This morphism induces a morphism on the associated long exact sequences on coho mology. The homotopy commutativity of the resulting diagram, after delooping once46o extend to the left, Z / Γ (cid:15) (cid:15) e β / / B Z U (1) β / / B Z immediately establishes the claim.Now it follows from expression (4–4) that d − Γ Sq ρ β is in the kernel of β . Byexactness of the Bockstein, this implies that it must be in the image of the exponentialmap, exp : H ∗ ( − ; R ) → H ∗ ( − ; U (1)). Hence there is an operation ψ : H ∗ ( − ; U (1)) → H ∗ + ( − ; R ) such that φ : = exp ◦ ψ = exp( ψ ) = d − Γ Sq ρ β . Equivalently, we have a factorization H ∗ ( − ; U (1)) ψ ' ' ❖❖❖❖❖❖❖❖❖❖❖ φ / / H ∗ + ( − ; U (1)) . H ∗ + ( − ; R ) exp ♠♠♠♠♠♠♠♠♠♠♠♠ Now the group of natural transformations H ∗ ( − ; U (1)) → H ∗ + ( − ; R ) is in bijectivecorrespondence with H ∗ + ( K ( U (1) , ∗ ); R ) ∼ =
0. Hence, ψ = Consequently,exp ◦ ψ =
0, so that φ =
0. Therefore, indeed we have d = Γ Sq ρ β . (cid:3) Remark 14
Theabove proposition suggests thatthese operations arerelated tosomesort of differential Steenrod squares. Indeed, this is the case, which has been investi gated by theauthors in[GS16a],with b Sq being one such operation.Now that we have established the algebraic construction, we turn to investigating theconvergence of the spectral sequence from a geometric point of view. In particular,we immediately observe that the only terms in the spectral sequence which contain In the published version, it was erroneously claimed that hom( H n ( M ; R ) , A ) ∼ = hom( H n ( M ; Q ) , A ) for any abelian group. This is false, e.g. for M = S n , A = Q , hom( R , Q )is a Q vector space of uncountable dimension, while hom( Q , Q ) ∼ = Q . The proof has beencorrected. q =
0. These terms converge to elementsin the filtered graded complex (since q = b K ( M ) / F b K ( M ) . Since the filtration is given by the ˇCech type filtration on M , we see that this quotientcontains elements which have nontrivial data on all open sets, intersections, and higherintersections. For the degrees q <
0, the filtration quotients F p b K ( M ) / F p + b K ( M )have trivial data below p intersections.In fact, it is not too surprising that this occurs. There is a geometric model forreduced b K which is given by the moduli stack ` n ∈ N B U ( n ) conn of unitary bundlesvector bundles, equipped with Hermetian connection. More precisely, let Vect ∇ bethe moduli stack of complex Hermetian vector bundles with Hermetian connections.It was shown in [BNV16] that, after taking the Grothendieck group completion, thereis a surjection given by the cycle mapcycl : Gr( π Vect ∇ ( M )) → b K ( M ) , which, in our construction, is equivalent tocycl : Gr (cid:16) π Map (cid:16) M , a n ∈ N B U ( n ) conn (cid:17)(cid:17) → b K ( M ) . Now the stack B U ( n ) conn can be identified with the moduli stack obtained by takingthe nerve of the action groupoid C ∞ ( − , U ( n )) // Ω ( − ; u ( n )) , with the action given bygauge transformations, where u is the Lie algebra of the unitary group. Let { U α } be agood open cover of M . Then a map M → ` n ∈ N B U ( n ) conn is given by the followingdata: ◦ A choice of smooth U ( n ) valued function g αβ on intersections U α ∩ U β . ◦ A choice of local connection 1 form A αβ on open sets U α .This is precisely the data needed to define a unitary vector bundle on M . Remark 15
More relevant to our needs though, is the fact that the effects of thefiltration become transparent when taking the completion of ` n ∈ N B U ( n ) conn as amodel for b K . We now see that the q = q < ifferential K theory. We now consider odd differential K theory K . In this casethe representing spectrum is the unitary group U itself. Viewing this as a classifyingspace we can write U = B Ω U . Of course we are interested in the correspondingstacks. Unfortunately, we do not have the analogue of the above group loop grouprelation in stacks, i.e. U conn B Ω U conn . Nevertheless, the machinery that we set upwill work equally well for differential K theory, as far as the third differential goes,i.e. d = b Sq still. However, the even differential are now transgressed in degree byone, so that they are also of odd degree. This is expected as the Chern character in thiscase is a map to cohomology of odd degree.The story for b K can be worked out similarly, as we indicated above. Let us expandon this in more details. In the odd case, the differential cohomology diamond takes theform Q k Ω k / im( d ) a ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ d / / Q k + Ω k + & & ▼▼▼▼▼▼▼▼▼▼▼▼▼▼ K R rrrrrrrrrrr ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆ b K I ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ R rrrrrrrrrrrr K R K U (1) ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ β K / / K ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ and we get a short exact sequence of presheaves (on the site of Cartesian spaces)0 → Z → Y k Ω k / im( d ) → b K → . It is straightforward to show that the map Z → Q k Ω k / im( d ) is zero. Consequently,we have the isomorphism b K ≃ Y k Ω k / im( d ) ≃ Y k + Ω k + . Using the same type of argument as in the even K theory K , we likewise get arefinement of the differential of the underlying topological theory. More precisely, wesee that the first nonzero differentials appear on the E page is490 Q k + Ω k + ( M ) − − H ( M ; U (1)) H ( M ; U (1)) − − H ( M ; U (1)) d b Sq Proposition 27
Proposition 26 holds for differential K theory. That is, the degreethreedifferential in b K isgivenbytherefinement oftheSteenrodsquare ofdimensionthree.Furthermore, using the same argument as in the proof of Proposition 25, we see that thepermanent cycles in bidegree (0 ,
0) are a subgroup of odd degree forms with rationalperiods.
Proposition 28
The group of permanent cycles in bidegree (0 ,
0) in the AHSS fordiff( Σ K , ch) is a subgroup of odd degree closed forms with rational periods. That is,wehave E , ∞ ⊂ Y k Ω k − , Q ( M ) ⊕ Z . Example 5 (Fields in string theory and M theory) InthestringtheoryandM theoryliteratureoneencounterssettingswherecohomologyclassesarecomparedtoK theoryelements, in the sense of asking when a cohomology class arises from or ‘lift to’ a K theory class. Thisinvolves, inasense, aphysical modelling oftheprocess of buildingtheAHSS.Onesuchobstruction is Sq ,viewedasthefirstnontrivial differential d inK theory, so that the condition Sq x = x amounts to sayingthat theclass liftto K theory. This isdesirable in thestudy ofthe partition function ofthe fields in type IIA string theory (see [DMW03] [KS03]). On the other hand, it is50esirabletohavedifferentialrefinementsforphysicalpurposes. Therefore,nowthatwehavethedifferential AHSSatourdisposal, itisnaturaltoconsider expressions suchas d ( b x ) : = b Sq ˆ x = b x that refines the topologicalclass x . This can be viewed as a condition on cohomology with U (1) coefficients (orflat n bundles), inorder that they liftto flatelements in b K . Ifthe degree of theclass x iseventhenweareintypeIIAstringtheoryandwelifttodifferential K theory. Onthe other hand, being in type IIB string theory means the degree of x is odd, and weareliftingtodifferential K theory. Thenewdifferentials d m and d m + arisingfromdifferential forms will correspond to even and odd degree closed differential forms,as the particular forms representing the physical fields F m and F m + via the Cherncharacter. Example 6 (D brane charges) The charges of D branes can a priori be taken to begiven as a class in cohomology Q H ∈ H ∗ ( X ; Q ). Quantum effects requires someof these charges to be (up to shifts) to be in integral cohomology. However, inorder to not discuss isomorphism classes of such physical objects but pinning downa particular physical object, one considers the charges to take values in differentialcohomology, with Deligne cohomology being one such presentation, Q ˆ H ∈ b H ∗ ( X ; Z )(see[CJM04]). Ontheotherhand,carefulanalysis revealsthatthecharges takevaluesin K theory rather than in cohomology Q K ∈ K i ( X ), for i = , Sq Q H =
0. Again at this stage adding in the geometry requires the chargestotakevaluesindifferential K theory Q b K ∈ b K i ( X ). Ourconstruction nowallowsforacharacterization of when charges in Deligne cohomology lift to charges indifferentialK theory,namely when they are annihilated by the third differential in the smoothAHSS,i.e. when b Sq Q ˆ H = There are various interesting generalized cohomology theories that descend from com plex cobordism, among which are Morava K theory and Morava E theory. Suchtheories can be defined using their coefficient rings, which in general are polynomialsover finite or p adic fields on generators whose dimension depends on the chromaticlevel and the prime p . As such, these kind of theories do not lend themselves directly This could end up being stronger in the sense that it is a condition for lifting differentialcohomology classes to differential K theory, but we will leave that for future investigations.
51o immediate geometric interpretation, in contrast to the case of K theory, which canbe formulated via stable isomorphism classes of vector bundles.However, recent work in [LSW16] (generalizing some aspects of [BDR04]) seemsto give hope in that direction. Nevertheless, just because an entity is defined overa finite field does not automatically make it ineligible for differential refinement. Infact, recently [GS16a] we have demonstrated this for the case of Steenrod cohomologyoperations, which are a priori Z / p valued operations. The main point there wasthat as long as these admit integral lifts then they do have a chance at a differentialrefinement. What we will seek here is something analogous: integral refinements ofsuch generalized cohomology theories.We will consider the integral Morava K theory e K ( n ), highlighted in [KS03] [Sa10][SW15]. Morava K theory K ( n ) is the mod p reduction of an integral (or p adic) lift e K ( n ) with coefficient ring e K ( n ) ∗ = Z p [ v n , v − n ]. This theory more closely resemblescomplex K theory than is the case for the mod p versions (for n =
1, it is the p completion of K theory). The integral theory is much more suited to applications inphysics [KS03] [Sa10] [Buh11] [SW15].The Atiyah Hirzebruch spectral sequence for Morava K theory has been studied byYagita in [Ya80] (see also [KS03]). There is a spectral sequence converging to K ( n ) ∗ ( X )with E term E p , q = H p ( X , K ( n ) q ). While this can be done for any prime, we willfocus on the prime 2. In this case, the first possibly nontrivial differential is d n + − ;this is given by [Ya80] d n + − ( xv kn ) = Q n ( x ) v k − n . Here Q n is the n th Milnor primitive at the prime 2, which may be defined inductivelyas Q = Sq , the Bockstein operation, and Q j + = Sq j Q j − Q j Sq j , where Sq j : H n ( X ; Z ) → H n + j ( X ; Z ) is the j th Steenrod square. These operations are derivations Q j ( xy ) = Q j ( x ) y + ( − | x | xQ j ( y ) . The signs are of course irrelevant at p =
2, but will become important in the inte gral version. Extensive discussion of the mod p Steenrod algebra in terms of theseoperations is given in [Ta99].The integral theory is also computable via an AHSS, which can be deduced from [KS03][SW15]. There is an AHSS converging to e K ( n ) ∗ ( X ) with E p , q = H p ( X , e K ( n ) q ). Thefirst possibly nontrivial differential is d n + − ; this is given by d n + − ( xv kn ) = e Q n ( x ) v k − n . Here e Q k : H ∗ ( X ; Z ) → H ∗ + k + − ( X ; Z ) is an integral cohomology operation lifting52he Milnor primitive Q k .In order to consider differential refinement of Morava K theory, we need geometricinformation encoded in differential forms, hence rational information. The rationaliza tion of Morava K theory e K ( n ), like any reasonable spectrum exists and can be thoughtof as localization at e K (0) = H Q . See [Bo79] [Ra84]. We can in the same way localizeat R . More precisely, the localized theory is given by e K R ( n ) = e K ( n ) ∧ M R , where M R is an Eilenberg Moore spectrum. We have an equivalence e K R ( n ) ≃ H (cid:0) Z [ v n , v − n ] ⊗ R (cid:1) and a Chern character mapch : e K ( n ) → H (cid:0) Z [ v n , v − n ] ⊗ Ω ∗ (cid:1) . Thus we can form the differential function spectrum diff( e K ( n ) , ch) and we can formthe associated AHSS. To see what form the spectral sequence takes, we need to discussthe flat Morava K theory e K U (1) ( n ), defined by the fiber sequence e K ( n ) → e K ( n ) ∧ M R → e K U (1) ( n ) : = e K ( n ) ∧ MU (1) . This theory is periodic with period 2(2 n − e K ( n ) and its rationalizationare periodic and we have a long exact sequence . . . e K ( n ) m ( M ) → ( e K ( n ) ∧ M R ) m ( M ) → e K mU (1) ( n )( M ) → e K ( n ) m + ( M ) → . . . relating the flat theory to both the rational ind integral theory. This, in particular, givesthe following identification. Lemma 29
Thecoefficients of flatMorava K theory are given by e K U (1) ( n ) m ( ∗ ) ≃ U (1) , m = n − , , otherwise . Knowing the coefficients of the flat theory, we can write down the relevant nonzero53erms on the E n − page of the corresponding spectral sequence Q k Ω k n − ( M ) ⊕ Z ... − n + + H n + − ( M ; U (1)) H n − ( M ; U (1))... − n + + H n − ( M ; R / Z ) d n − and the only nonzero differential is given by d n − : Y k Ω k n − ( M ) ⊕ Z → H n − ( M ; R / Z ) . Just as in the case for differential K theory (see Propositions 25 and 28), we have thefollowing. Proposition 30
The group of permanent cycles in bidegree (0 ,
0) in the AHSS fordiff( e K ( n ) , ch) is a subgroup of certain closed forms with rational periods. Moreprecisely, wehave E , ∞ ⊂ Y k Ω k (2 n − , Q ( M ) ⊕ Z . To identify the the ˇCech cohomology groups with coefficients in b K ( n ) , we make theidentification (as we did for differential K theory) b K ( n ) ≃ Y k Ω k (2 n − ⊕ Z on the site of Cartesian spaces. Again, using the sheaf condition over smooth manifolds,54e have H p ( M ; b K ( n ) ) ≃ Y k Ω k (2 n − ( M ) ⊕ Z . We now consider the differential refinement of the (integrally lifted) Milnor primitive.As before, let Γ : H n ( − ; Z / → H n ( − ; U (1)) denote the map induced by therepresentation of Z / ρ : Z → Z / Lemma 31
The integral Milnor primitive e Q n factors through the representation Γ : Z / ֒ → U (1). Thatis,there exists anoperation b Q n such that Q n ρ = ρ e Q n = ρ β Γ b Q n , where β isthe Bockstein for the exponential sequence. Proof.
Recall first that ρ β Γ = ρ ˜ β = Sq , where ˜ β is the Bockstein for the mod 2reduction sequence. We can therefore rewrite the above equation as Q n ρ = ρ e Q n = ρ β Γ b Q n = Sq b Q n . and the existence of the class b Q n holds if and only if Sq Q n ρ =
0. On the other hand,the existence of the integral lift e Q n immediately implies this condition. (cid:3) Again, let β and ˜ β denote the Bockstein homomorphism corresponding to the se quences 0 → Z → R → R / Z → → Z → Z → Z / →
0, respectively. Thenthe following can be proved in a similar way as we did for Proposition 26 in the caseof differential K theory.
Proposition 32 (Odd differentials for Morava AHSS) The (2 n + −
1) differential inthe AHSSfordifferential Morava K theory isgiven by d n + − = Γ b Q n ρ β, q < , e Q n , q > , , q = . Remark 16 (Odd primes) Theabovediscussion hasbeenfortheprime2,thatis,weare considering integral Morava K theory as arising from lifting of the p = p . A similar discussion follows and we have an integral liftoftheMilnor primitiveatoddprimes,asinLemma31. Thedifferentials willbeagaingivenbytheserefinementoftheMilnorprimitive,i.e. Proposition 32holdsexceptthatthe primitives are defined using the Steenrod reduced power operations P j . Precisely, Q is the Bockstein homomorphism associated to reduction mod p sequence, andinductively Q i + = P p i Q i − Q i P p i . Theoperations P j have been differentially refinedin [GS16a]. Hence the refinement of the Milnor primitives at odd primes will alsofollow. Thenthe ( p n + −
1) differential intheAHSSfordifferential MoravaK theoryisgiven by d p n + − = Γ p b Q n ρ p β, q < , e Q n , q > , , q = . Example 7 (Lifting fields to differential Morava K theory) WewillbuildonExample5 and aim to lift the cohomology classes beyond K theory. In particular, for x = λ = p the first Spin characteristic class, wehave b x = ˆ λ the differential refinements of λ [SSS12] [FSSt12] (which can be viewed as a lifted Wu class [HS05]) we would have b Sq ˆ λ =
0. Thiscondition indifferential cohomology canbeviewedasarefinementofthe condition W = Sq λ = p =
2) as shown in [KS03] and elaborated furtherin[Buh11]. Fromthestructure ofthesmooth AHSSinrelation totheclassical AHSS,onecanextendvariousresultstothedifferential case. Forinstance, onecangeneralizethestatementin[KS03]onorientationtostatethat: an oriented smooth 10 dimensionalmanifold is oriented with respect to differential (integrally lifted from p =
2) MoravaK(2) theory b K (2) if the class ˆ W : = b Sq ˆ λ =
0. Thedevelopment ofthisaswellastherelation to refinements of characteristic classes deserves a separate treatment and willbeaddressed elsewhere.
Remark 17 (i)
NotethatourconstructionallowsforanAHSSforotherspectrabeyondthe particular ones we discussed above. This holds for any spectrum which admits arationalization, whosecoefficients areknown,andwhichcanbeliftedintegrally inthesense that wediscussed at the beginning ofthis section. (ii)
All the cohomology theories that we used in this paper can be twisted. Indeed,the construction in this paper can be generalized to construct an AHSS for twisteddifferential spectra [GS16b],inthe sense of[BN14].56 cknowledgement
The authors would like to thank Ulrich Bunke, Thomas Nikolaus, and Craig Westerlandfor interesting discussions at the early stages of this project and Urs Schreiber for veryuseful comments on the first draft. We are grateful to the anonymous referee for acareful reading and for useful suggestions.
References [Ad74] J. F. Adams,
Stable homotopy and generalised cohomology , The Univ. of ChicagoPress, Chicago, 1974.[AH62a] M. F. Atiyah and F. Hirzebruch,
Vector bundles and homogeneous spaces , 1961Proc. Sympos. Pure Math. vol. III, pp. 7–38, American Math. Soc., Providence, R.I.,1962.[AH62b] M. F. Atiyah and F. Hirzebruch,
Analytic cycles on complex manifolds , Topology (1962), 25–45.[Ar92] D. Arlettaz, The order of the differentials in the Atiyah Hirzebruch spectral sequence , K Theory (1992), no. 4, 347–361.[BDR04] N. A. Baas, B. I. Dundas, and J. Rognes, Two vector bundles and forms of ellipticcohomology , Topology, geometry and quantum field theory, London Math. Soc. LectureNote Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 18–45.[Bj95] A. Bj¨orner,
Topological Methods , Handbook of Combinatorics, North Holland, Ams terdam, 1995, pp. 1819–1872.[Bo79] A. K. Bousfield,
The localization of spectra with respect to homology , Topology (1979), no. 4, 257–281.[BT82] R. Bott and L. W. Tu, Differential forms in algebraic topology , Springer Verlag, NewYork Berlin, 1982.[BMRS08] J. Brodzki, V. Mathai, J. Rosenberg, and R. J. Szabo,
D Branes, RR fields andduality on noncommutative manifolds , Commun. Math. Phys. (2008), 643–706,[ arXiv:hep-th/0607020 ].[Br73] K. Brown,
Abstract homotopy theory and generalized sheaf cohomology , Trans. Amer.Math. Soc. (1973), 419–458 .[Br93] J L Brylinski,
Loop spaces, characteristic classes and geometric quantization ,Progress in Math. 107, Birkh¨auser, Boston, 1993.[BM96] J L Brylinski and D. A. McLaughlin, ˇCech cocycles for characteristic classes , Com munications Math. Phys. (1996), 225–236.[Buh11] L. Buhn´e,
Properties of integral Morava K theory and the asserted application tothe Diaconescu Moore Witten anomaly , Diploma thesis, Hamburg University, 2011. Bu12] U. Bunke,
Differential cohomology , [ arXiv:math.AT/1208.3961 ].[BKS10] U. Bunke, M. Kreck, and T. Schick,
A geometric description of differential coho mology , Ann. Math. Blaise Pascal (2010), no. 1, 1–16.[BN14] U. Bunke and T. Nikolaus, Twisted differential cohomology , [ arXiv:1406.3231 ].[BNV16] U. Bunke, T. Nikolaus, and M. V¨olkl,
Differential cohomology theories as sheavesof spectra , J. Homotopy Relat. Struct. (2016), no. 1, 1–66.[BS09] U. Bunke and T. Schick, Smooth K theory , Ast´erisque (2009), 45 135 (2010)[BS10] U. Bunke and T. Schick,
Uniqueness of smooth extensions of generalized cohomologytheories , J. Topol. (2010) 110–156.[BSSW09] U Bunke, T Schick, I Schr¨oder, and M Wiethaup, Landweber exact formal grouplaws and smooth cohomology theories , Algebr. Geom. Topol. (2009) 1751–1790.[CJM04] A. L. Carey, S. Johnson, and M. K. Murray, Holonomy on D branes , J. Geom. Phys. (2004), 186–216, [ arXiv:hep-th/0204199 ].[CS85] J. Cheeger and J. Simons, Differential characters and geometric invariants , volume1167 of Lecture Notes in Math., pages 50–80. Springer, Berlin, 1985.[DMW03] E. Diaconescu, G. Moore, and E. Witten, E gauge theory, and a deriva tion of K theory from M theory, Adv. Theor. Math. Phys. (2003) 1031–1134,[ arXiv:hep-th/0005090 ].[Do62] A. Dold, Relations between ordinary and extraordinary cohomology , Colloq. Algebr.Topology, Aarhus, 1962, 2–9.[Du01] D. Dugger,
Combinatorial model categories have presentations , Adv. Math. (2001), no. 1, 177–201, [ arXiv:math/0007068 ] [ math.AT ].[DL05] J. L. Dupont and R. Ljungmann,
Integration of simplicial forms and Deligne coho mology , Math. Scand. (2005), no. 1, p. 11–39.[EV92] H. Esnault and E. Viehweg, Lectures on Vanishing Theorems , Birkh¨auser, DMVSeminar 20, 1992.[FSS13] D. Fiorenza, H. Sati, and U. Schreiber,
Extended higher cup product Chern Simonstheory , J. Geom. Phys. (2013), 130–163, [ arXiv:1207.5449 ] [ hep-th ].[FSS15a] D. Fiorenza, H. Sati, and U. Schreiber, A Higher stacky perspective on Chern Simons theory , Mathematical Aspects of Quantum Field Theories (Damien Calaqueand Thomas Strobl eds.), Springer, Berlin (2015), [ arXiv:1301.2580 ] [ hep-th ].[FSSt12] D. Fiorenza, U. Schreiber, and J. Stasheff, ˇCech cocycles for differential charac teristic classes – An infinity Lie theoretic construction , Adv. Theor. Math. Phys. (2012), 149–250,[ arXiv:1011.4735 ] [ math.AT ].[FFG86] A. T. Fomenko, D. B. Fuchs, and V. L. Gutenmacher, Homotopic Topology ,Akam´emiai Kiad´o, Budapest, 1986. Fr00] D. S. Freed,
Dirac charge quantization and generalized differential cohomology , Surv.Diff. Geom., VII, pages 129–194, Int. Press, Somerville, MA, 2000.[FL10] D. S. Freed and J. Lott,
An index theorem in differential K theory , Geom. & Top. (2010), 903–966.[FW99] D. S. Freed and E. Witten, Anomalies in string theory with D branes , Asian J. Math. (1999), 819, [ arXiv:hep-th/9907189 ].[Ga97] P Gajer, Geometry of Deligne cohomology , Invent. Math. (1997) 155–207.[GL15] A. Gorokhovsky and J. Lott,
A Hilbert bundle description of differential K theory ,[ arXiv:1512.07185 ] [math.DG].[GS15] D. Grady and H. Sati, Massey products in differential cohomology via stacks ,[ arXiv:1510.06366 ] [math.AT].[GS16a] D. Grady and H. Sati, Primary operations in differential cohomology ,[ arXiv:1604.05988 ] [math.AT].[GS16b] D. Grady and H. Sati, AHSS for twisted differential spectra , to appear.[GM81] P.A. Griffiths and J.W. Morgan,
Rational homotopy theory and differential forms ,Progress in Mathematics, 16, Birkh¨auser, Boston, Mass., 1981.[Ha02] A. Hatcher,
Algebraic topology , Cambridge University Press, Cambridge, 2002.[HMSV15] P. Hekmati, M. K. Murray, V. S. Schlegel, and R. F. Vozzo,
A Geometric model forodd differential K theory , Diff. Geom. Appl. (2015), 123–158, [ arXiv:1309.2834 ][math.KT].[Hi71] P. J. Hilton, General Cohomology and K theory , Cambridge University Press, Cam bridge, 1971.[Ho12] M. H. Ho,
The differential analytic index in Simons Sullivan differential K theory ,Ann. Global Anal. Geom. (2012), 523–535, [ arXiv:1110.0151 ] [math.DG].[Ho14] M. H. Ho, Remarks on flat and differential K theory , Ann. Math. Blaise Pascal (2014), 91–101, [ arXiv:1203.5383 ] [math.DG].[HS05] M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, andM theory , J. Differential Geom. (3) (2005), 329–452.[HJJS08] D. Husem¨oller, M. Joachim, B. Jurco, and M. Schottenloher, Basic Bundle Theoryand K Cohomology Invariants , Springer, Berlin, 2008.[Ja87] J. F. Jardine,
Stable homotopy theory of simplicial presheaves , Canadian J. Math. (1987), 733–747.[Ja15] J. F. Jardine, Local Homotopy Theory , Springer Verlag, New York, 2015.[Ka87] M. Karoubi,
Homologie cyclique et K th´eorie , Ast´erisque no. 149 (1987).[Kl08] K. R. Klonoff,
An index theorem in differential K theory , PhD Thesis, The Universityof Texas at Austin, 2008. KS03] I. Kriz and H. Sati,
M theory, type IIA superstrings, and elliptic cohomology , Adv.Theor. Math. Phys. (2004) 345, [ arXiv:hep-th/0404013 ].[LSW16] J. A. Lind, H. Sati, and C. Westerland, A higher categorical analogue of topologicalT duality for sphere bundles , [ arXiv:1601.06285 ] [math.AT].[Lo94] J. Lott, R / Z index theory , Comm. Anal. Geom. (1994), no. 2, 279 311.[Lo00] J. Lott, Secondary analytic indices , Progr. Math., 171, 231–293, Birkh¨auser Boston,Boston, MA, 2000.[Lu10] J. Lurie, Course notes on
Chromatic Homotopy Theory , lecture 25 (2010), ∼ lurie/252x.html [Lu11] J. Lurie, Higher algebra , prepublication book draft, available at , 2011.[Ma63] C. R. F. Maunder,
The spectral sequence of an extraordinary cohomology theory ,Proc. Cambridge Philos. Soc. (1963), 567–574.[MS06] J. P. May and J. Sigurdsson, Parametrized Homotopy Theory , Amer. Math. Soc.,Providence, RI, 2006.[Mc01] J. McCleary,
A User’s Guide to Spectral Sequences , 2nd ed., Cambridge UniversityPress, Cambridge, 2001.[MM97] R. Minasian and G. Moore,
K theory and Ramond Ramond charge , J. High EnergyPhys. (1997), 002, [ arXiv:hep-th/9710230 ].[Mi81] H. R. Miller,
On relations between Adams spectral sequences, with an application tothe stable homotopy of a Moore space , J. Pure Appl. Algebra (1981), no. 3, 287–312.[Pr06] V. V. Prasolov, Elements of Combinatorial and Differential Topology , Amer. Math.Soc., Providence, RI, 2006.[Ra84] D. C. Ravenel,
Localization with respect to certain periodic homology theories , Amer.J. Math. (1984), no. 2, 351–414.[Ru08] Y. Rudyak,
On Thom Spectra, Orientability and Cobordism , Springer Verlag, Berlin,2008.[Sa10] H. Sati,
Geometric and topological structures related to M branes , In Superstrings,geometry, topology, and C ∗ algebras, 181–236, Proc. Sympos. Pure Math., 81, Amer.Math. Soc., Providence, RI, 2010, [ arXiv:1001.5020 ] [math.DG].[SSS12] H. Sati, U. Schreiber, and J. Stasheff, Differential twisted String and Fivebrane struc tures , Commun. Math. Phys. (2012), 169–213, [ arXiv:0910.4001 ] [ math.AT ].[SW15] H. Sati and C. Westerland,
Twisted Morava K theory and E theory , J. Topol. (2015),no. 4, 887–916, [ arXiv:1109.3867 ] [math.AT].[Sc13] U. Schreiber, Differential cohomology in a cohesive infinity topos ,[ arXiv:1310.7930 ] [math ph].[Sch] S. Schwede, Symmetric Spectra , book draft Sh07] B. Shipley. H Z algebra spectra are differential graded algebras , Amer. J. Math. (2) (2007), 351–379.[SS08] J. Simons and D. Sullivan, Axiomatic characterization of ordinary differential coho mology
J. Topol. (1) (2008), 45–56.[Sw75] R. Switzer, Algebraic Topology , Springer Verlag, Berlin, 1975.[Ta99] H. Tamanoi, L subalgebras, Milnor basis, and cohomology of Eilenberg MacLanespaces , J. Pure Applied Alg. (1999), 153–198.[TWZ13] T. Tradler, S. O. Wilson, and M. Zeinalian, An elementary differential extension ofodd K theory , J. K Theory (2013), no. 2, 331 361.[TWZ15] T. Tradler, S. O. Wilson, and M. Zeinalian, Differential K theory as equivalenceclasses of maps to Grassmannians and unitary groups , [ arXiv:1507.01770 ].[Up14] M. Upmeier,
Refinements of the Chern Dold character: Cocycle additions in differ ential cohomology , [ arXiv:1404.2027 ] [math.AT].[Up15] M. Upmeier,
Algebraic structure and integration maps in cocycle models for differ ential cohomology , Algebr. Geom. Topol. (2015), no. 1, 65–83.[Ya80] N. Yagita, On the Steenrod algebra of Morava K theory , J. London Math. Soc. (3)(1980), 423–438.DepartmentofMathematics,NewYorkUniversityAbuDhabi,AbuDhabi,UAEDepartmentofMathematics,NewYorkUniversityAbuDhabi,AbuDhabi,UAEDepartmentofMathematics,UniversityofPittsburghPittsburgh,PA15260,USA [email protected], [email protected]@nyu.edu, [email protected]