Higher-twisted periodic smooth Deligne cohomology
aa r X i v : . [ m a t h . DG ] F e b HIGHER-TWISTED PERIODIC SMOOTH DELIGNE COHOMOLOGY
DANIEL GRADY AND HISHAM SATI
Abstract.
Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology,was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rhamcohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomologywe need a periodic version. Combining the periodic versions of both ingredients leads us to introduce aperiodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe ofany odd degree. We present the main properties of the new theory and illustrate its use with examples andcomputations, mainly via a corresponding twisted differential Atiyah-Hirzebruch spectral sequence.
Contents
1. Introduction 12. Twisted periodic integral cohomology 52.1. Twists via bundles of spectra 52.2. Properties of twisted periodic integral cohomology 103. Periodic smooth Deligne cohomology 113.1. Construction as a cohomology theory 113.2. Ring structure and examples 134. Twisted periodic smooth Deligne cohomology 144.1. The parametrized spectrum and gerbes via smooth stacks 144.2. Properties of twisted periodic smooth Deligne cohomology 205. The spectral sequence { AHSS h and examples 215.1. The spectral sequence in twisted periodic smooth Deligne cohomology 215.2. Examples via the spectral sequence 23References 241. Introduction
There has been a lot of recent activity on modifying generalized cohomology theories to include twists andgeometric refinements, in order to account for automorphisms and include geometric data. Twisted differen-tial generalized cohomology theories are established at the general axiomatic level [BN14]. However, workingout these theories explicitly is in practice not a straightforward task. Twisting the simplest case of a differ-ential cohomology theory, namely Deligne cohomology, proved to be nontrivial [GS17a] and is closely relatedto interesting constructions in algebraic geometry, namely taking coefficients in variations of mixed Hodgestructures (see [CH89][Ha15]). Even at the topological level, while twisting of generalized cohomology theo-ries is axiomatically well-established [MS06][ABGHR14][ABG10], spelling out explicit constructions requiresconsiderable work (see [ABG10][SW15][LSW16] for recent illustrations). The goal of this paper is to general-ize the degree one twists of Deligne cohomology from [GS17a] to include twists of higher degrees. These willbe in the form of higher gerbes, or n -bundles, with connections (see [FSSt12][SSS12][Sc13][FSS13][FSS15a]for constructions and related applications).Deligne cohomology (see [De71][Be85][Gi84][Ja88][EV88][Ga97]) is a differential refinement of ordinary, i.e.integral, cohomology. As such it has various realizations (see [De71][CS85][Ga97][Br93][DL05][HS05][BKS10] BB14][Sc13]), which are (expected to be) equivalent (see [SS08][BS10]). Consider the sheaf of chain com-plexes associated with real-valued differential forms (1.1) D p n q : “ ` . . . / / / / Z / / Ω d / / Ω d / / . . . d / / Ω n ´ ˘ , where we place differential p n ´ q -forms in degree 0 and locally constant integer-valued functions in degree n . Given a smooth manifold M , the Deligne cohomology group of degree n is defined to be the sheaf(hyper-)cohomology group p H n p M ; Z q : “ H p M ; D p n qq . ˇCech resolutions allow for explicit calculation ofthese groups. If t U α u is a good open cover of M , then one can form the ˇCech-Deligne double complex (see[BT82][Br93])(1.2) ś α , ¨¨¨ ,α n Z p U α ...α n q / / ś α , ¨¨¨ ,α n Ω p U α ...α n q d / / ś α , ¨¨¨ ,α n Ω p U α ...α n q d / / . . . d / / ś α , ¨¨¨ ,α n Ω n ´ p U α ...α n q ś α , ¨¨¨ ,α n ´ Z p U α ...α n ´ q / / p´ q n ´ δ O O ś α , ¨¨¨ ,α n ´ Ω p U α ...α n ´ q d / / p´ q n ´ δ O O ś α , ¨¨¨ ,α n ´ Ω p U α ...α n ´ q d / / p´ q n ´ δ O O . . . d / / ś α , ¨¨¨ ,α n ´ Ω n ´ p U α ...α n ´ q p´ q n ´ δ O O ... p´ q n ´ δ O O ... p´ q n ´ δ O O ... p´ q n ´ δ O O ... p´ q n ´ δ O O ś α ,α Z p U α α q / / ´ δ O O ś α ,α Ω p U α α q d / / ´ δ O O ś α ,α Ω p U α α q d / / ´ δ O O . . . d / / ś α ,α Ω n ´ p U α α q ´ δ O O ś α Z p U α q / / δ O O ś α Ω p U α q d / / δ O O ś α Ω p U α q d / / δ O O . . . d / / ś α Ω n ´ p U α q , δ O O where U α α ...α k denotes the k -fold intersection U α X U α X ¨ ¨ ¨ X U α k . With d and δ the de Rham and ˇCechdifferentials, respectively, acting on elements of degree p , the total operator on the double complex is theˇCech-Deligne operator D : “ d ` p´ q p δ . The sheaf cohomology group H p M ; D p n qq can then be identifiedwith the group of diagonal elements η k,k in the double complex which are D -closed, Dη k,k “
0, modulo thosewhich are D -exact.At a more general level, and from a homotopy theory point of view, given a spectrum E the canonicaldata for the corresponding differential theory is comprised of the following (see [Bu12, Example 4.49]). Let H be the Eilenberg-MacLane functor and take A : “ π ˚ E b R the ‘realified’ coefficients of the theory. Let c : E Ñ H p A q be the map uniquely determined up to homotopy such that it induces the map ‘realifying’ thecoefficients π ˚ p E q Ñ π ˚ p A q – π ˚ p E q b R , x ÞÑ x b
1. Indeed, for E “ H p Z q the integral Eilenberg-MacLanespectrum, A “ Z b R – R , and c : H p Z q Ñ H p R q uniquely determined by Z Ñ R – Z b R , x ÞÑ x b
1. Thisdata determines a differential extension of H Z , which in degree n takes the form p Σ n H Z , R , c q . Applying theEilenberg-MacLane functor H to the Deligne complex D p n q from expression (1.1) gives a natural equivalenceof differential spectra (see [Bu12]) H p D p n qq – p Σ n H Z , R , c q . What would it mean to twist Deligne cohomology? At a practical level, Deligne cohomology is a fusion ofde Rham cohomology and ˇCech cohomology. One could naively consider twists of de Rham cohomology, madeperiodic, as well as twists of ˇCech cohomology, separately. Now, the ˇCech-Deligne operator D “ d ` p´ q p δ acting on the ˇCech-Deligne double complex is a combination of the de Rham differential d , acting in thedirection of the de Rham complex, and the ˇCech differential δ , acting in the direction of the ˇCech complex(see again [BT82][Br93] for extensive discussions). One could then consider two situations (or stages) in This is sometimes also denoted Z D p n q or Z p n q D . We are in the smooth setting throughout, so we will not need extradecorations. This would be H n p M ; D p n qq if we use the opposite convention. However, the one we use is positively graded, hence betteradapted for stacks. ttempting to twist the operator D : First, adding a closed differential form H , i.e. modifying only the formpart, leading to twisting of the de Rham differential d ❀ d H . Second, adding a differential cohomologyclass ˆ h , i.e. modifying both parts of the whole ˇCech-de Rham differential, i.e. d ❀ d H and δ ❀ δ B where p H, B q are appropriate (de Rham, ˇCech)-components of ˆ h . We will see that it will not quite work that way;nevertheless, this turns out to be a good heuristic to keep in mind. In contrast to the de Rham differential,twisting the ˇCech differential would be quite complicated due to higher and higher local transition data.Indeed, ˇCech-de Rham cohomology, twisted by a degree three form (without altering the ˇCech direction) isused in [GT10] for describing a finite-dimensional model of twisted K-theory. Note that twisted de Rhamcohomology of a space can be described via the untwisted cohomology of a corresponding stack [BSS07].Which degrees should the twists H or ˆ h have? The twists of the Deligne complex, a priori naturallyarise in degree one [GS17a]. Note that for the underlying topological theory, a representation of the fun-damental group π p X q of a space X on Aut p Z q – Z { Z the structure of a module over the groupring Z r π p X qs , which is used in [BFGM03] to describe π p X q -twisted integral cohomology. On the otherhand, one can twist the de Rham complex by differential forms of any odd degree, not just degree one(see [RW86][BCMMS02][Te04][Sa09][Sa10][MW11]). At first glance, this might appear to give an inherentincompatibility of twisted de Rham cohomology and twisted integral cohomology. However, if one takes acloser look, one realizes that twisted de Rham cohomology is really about Z { periodic integral cohomology. Hence we consider twists of the latter theory in Section 2. This then paves part of theway for us to go towards a general twisted Deligne cohomology. However, as both ingredients, namely deRham and integral cohomology, were made periodic, we define a periodic version of Deligne cohomology inSection 3. We characterize its main properties via sheaf cohomology and differential spectra, including thering structure arising from the Deligne-Beilinson cup product [De71][Be85] (see [FSS13][FSS15a]). Periodicintegral differential cohomology groups p H p X ; Z r u, u ´ sq have been considered from an index theoretic pointof view briefly in [Lo02][FL10, Sec. 8.4].Having defined the appropriate starting point for the twisting of Deligne cohomology, namely the periodicversion, we discuss the twists of periodic Deligne cohomology in Section 4. We approach twisting of periodicDeligne cohomology using simplical presheaves and smooth stacks [FSSt12][FSS13][HQ15][FSS15a][Sc13], aswe did in [GS17a]. This approach is very well-suited to the higher twists and allows for the use of powerfulalgebraic machinery. We will show that the twists indeed refine the twists of both integral cohomology andthe de Rham complex. Smooth stacks will arise naturally in twisting periodic Deligne cohomology. Justas we can twist periodic integral cohomology by odd degree singular cocycles (Section 2), we will see thatperiodic Deligne cohomology can be twisted by higher gerbes of odd degree (Section 4). The appearance ofgerbes naturally leads us into the world of smooth stacks, and we will find it useful to recall some of theconstructions in this setting (see [Br93][FSSt12][FSS13][FSS15a]). This requires us to understand in detailexactly what we mean by twisting a periodic differential cohomology theory. We give a characterization ofthe twists via moduli stacks of higher bundles with connections, but keeping technical matters to a minimum.As discussed above, we can twist Deligne cohomology by gerbes of odd degree. It is interesting to see wherethe gerbe data appears in defining the twisted theory. In fact, as observed in [BN14], a crucial ingredientin defining twisted differential theories is the analogue of the de Rham isomorphism theorem for twistedcohomology. In the untwisted case, recall that the locally constant sheaf R admits an acyclic resolution viathe de Rham complex(1.3) R (cid:31) (cid:127) / / Ω d / / Ω d / / Ω / / . . . , and the de Rham Theorem is manifestly a corollary of this fact. Indeed, for a smooth manifold M , the sheafcohomology H ˚ p M, R q can be calculated both as ˇCech cohomology and via this resolution. The isomorphismbetween singular and ˇCech cohomology then recovers de Rham’s classical theorem.Just as multiplicative cohomology theories have topological spaces of twists (the Picard spaces) [MQRT77][MS06][ABGHR14], differential refinements of such theories have smooth stacks of twists. Indeed, the stackof twists y Tw p R for any differentially refined cohomology theory p R “ p R , c, A q was introduced in [BN14]. This as defined by the pullback (in the notation of [GS17b])(1.4) y Tw p R / / (cid:15) (cid:15) Pic form R (cid:15) (cid:15) Pic top R / / Pic dR R , where ‚ Pic top R is the ordinary Picard -groupoid of twists for the ring spectrum R , embedded as a constantsmooth stack, ‚ Pic dR R is the Picard stack of sheaves of invertible module spectra over the smash product R ^ H R (embedded as a constant sheaf of spectra), and ‚ Pic form R is the smooth stack which (after evaluation on a smooth manifold M ) comes as the nerve ofthe groupoid whose objects are weakly locally constant, K-flat, invertible modules over Ω ˚ p´ ; A q| M (see [BN14] for details).An element of the pullback (1.4) can be identified with a triple p R ˆ τ “ p R τ , t, L q , where R τ is an underlyingtwisted cohomology theory with a topological twist τ , L is an invertible module over Ω ˚ p´ ; A q and t is anequivalence t : R τ ^ H R » ÝÑ H p L q , exhibiting a twisted de Rham theorem. This stack will be important in identifying the twists for periodicDeligne cohomology in Section 4.The situation is summarized in the following tables. The first one captures the untwisted case Untwisted cohomology
Ordinary PeriodicUnderlying theory
Locally constant sheaf R Sheaf of graded algebras R r u, u ´ s de Rham complex Ordinary de Rham complex Ω ˚ Periodic complex Ω ˚ r u, u ´ s In [GS17b] we highlighted the close analogies between twisted spectra and line bundles, in that twisteddifferential spectra are closely related to bundles of spectra equipped with a flat connection. Here, we wishto replace the notions in the above table with the twisted analogues as follows.
Twisted cohomology
Ordinary PeriodicUnderlying theory
Locally constant sheaf L Sheaf of DGA-modules L ‚ Twist degree
One Any odd degree
Geometric twisting object
Line bundle with flat connection d ` H Gerbe with curvature H k ` de Rham complex (Ω ˚ b L , d ` H ^ ) (Ω ˚ b L ‚ , d ` H k ` ^ )The theories that we consider are related schematically as follows(1.5) Periodic Deligne p H ˚ p M ; D r u, u ´ sq forgetconnection (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O forgetperiodicity / / /o/o/o/o/o/o/o/o/o/o/o Deligne p H ˚ p M ; Z q “ H p M ; D p˚qq forgetconnection (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O Periodic integral H ˚ p M ; Z r u, u ´ sq forgetperiodicity / / /o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o Integral H ˚ p M ; Z q where the top row, bottom row, left column and right column represents geometric theories, topologicaltheories, periodic theories, and non-periodic theories, respectively. The relations between the corresponding We will be dealing with p8 , q -categories, so that whenever we talk about pullbacks, pushouts, or any other universalconstruction, we mean it in the p8 , q -sense, i.e., up to higher coherence homotopy. Whenever we draw a diagram, it shouldbe understood, whether we draw it explicitly or not, that there are homotopies and higher homotopies involved, up to theappropriate degree! paces of twists are in turn summarized in the schematic diagram(1.6) B p Z { q ∇ ˆ ś k ą B k U p q ∇ | ¨ | (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O u “ / / /o/o/o/o/o/o/o/o/o/o/o/o B p Z { q ∇ | ¨ | (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O K p Z { , q ˆ ś k ą K p Z , k ` q u “ / / /o/o/o/o/o/o/o/o/o/o/o K p Z { , q . Here | ¨ | is geometric realization, which reduces a geometric theory down to the corresponding topologicaltheory, and B p Z { q ∇ is the stack of twists for Deligne cohomology [GS17a]. To twist the theories displayedin the first schematic diagram (1.5) one would consider maps from the manifold M to the correspondingspace of twists in the second schematic diagram (1.6).Explicit ˇCech cocycles for Deligne cohomology are described in [Ga97][BM94][BM96][Go06]. While we donot do this in full generality in the twisted case, we do explain how the ˇCech cocycle data appear as part of thetrivializing data for twisted periodic Deligne cohomology in Remark 6. This involves Chern-Simons type triv-ialization of ˇCech-Deligne cocycles, packaged succinctly as in [GS17b]. Extensive discussions of such trivial-izations relating to Chern-Simons theory are given in [BM94][BM96][Go01][Fr02][CJMSW05][FSSt12][Wa13][GPT13][FSS14][Sa14][FSS15a][FSS15b][Th15]. In contrast, cocycles arising from chain complexes would in-volve an abstract higher local system resulting from the twists of periodic integral cohomology. While this isdoable, it does not make the description any more transparent in comparison to the description via spectra;hence we do not consider it in this paper.Making use of the general constructions in [GS16b][GS17b], we then consider the Atiyah-Hirzebruchspectral sequence for twisted periodic integral cohomology as well as for twisted periodic Deligne cohomologyin Section 5. We provide explicit constructions and characterizations in Section 5.1 and then illustrate thecomputations via examples in Section 5.2.We note that there are other approaches to studying the ˇCech-de Rham double complex. Explicit de-scription of cocycles via the cohomology of the total operator D of the double complex is provided in [Pi05].Using the notion of Cheeger-Simons cochain sparks [CS85], a homological machine for the study of secondarygeometric invariants called spark complexes is described in [HLZ03] [HL06]. This seems to be an appropriatesetting for twisting differential cohomology in its incarnation as differential characters. While we do notaddress this, we expect that the resulting twisted versions would be equivalent; our approach places thecomplication in the coefficients of the (hyper)cohomology while that more homologically flavored approachwould place it in the cycles, e.g., via spark complexes.2. Twisted periodic integral cohomology
In this section, we describe the twisted periodic cohomology with both real and integral coefficients. Thisgeneralizes twists of integral cohomology [MQRT77], also described in modern categorical terms in [ABG10]and geometrically in [Fr01]. This will be a precursor for the de Rham theorem needed to define twistedDeligne cohomology. Throughout the remainder of this paper, we will follow the -categorical treatment oftwisted cohomology theories [ABG10][ABGHR14][SW15] and their differential refinements [BN14][GS17b].2.1. Twists via bundles of spectra.
The starting point for periodic integral cohomology is the differentialgraded algebra (DGA) Z r u, u ´ s , equipped with the trivial differential. There is a functor(2.1) H : C h ÝÑ S p , from the category of unbounded chain complexes to the category of spectra, called the Eilenberg-MacLanefunctor. This functor was defined in [Sh07], where it was shown to exhibit an equivalence between H Z -module spectra and differentially graded Z -algebras. Applying H to Z r u, u ´ s we get a spectrum H Z r u, u ´ s which represents periodic integral cohomology, in the sense that(2.2) H ˚ p X ; Z r u, u ´ sq – H ˚ p X ; Z qr u, u ´ s , where the right hand side is the graded algebra whose elements are formal Laurent polynomials with coeffi-cients in H ˚ p X ; Z q graded by homogeneous degree. The ring structure on the right is induced from the cupproduct structure on H ˚ p X ; Z q , while on the left it is induced from the algebra structure on Z r u, u ´ s . This heory is naturally Z { H ˚ p X ; Z r u, u ´ sq – H ˚` p X ; Z r u, u ´ sq .For this reason, we will usually refer to the degree of a class as either even or odd . Remark 1 (Action of units on periodic integral cohomology) . Being an H Z -module spectrum, the spectrum H Z r u, u ´ s receives an action by H Z . This action manifests itself simply by the action of the cup product inintegral cohomology. More precisely, given a cohomology class h P H ˚ p X ; Z q , we can act on a homogeneousLaurent polynomial of degree k , a “ ¨ ¨ ¨ ` a k ` u ´ ` a k ` a k ´ u ` ¨ ¨ ¨ , via p h, a q ÞÝÑ ¨ ¨ ¨ ` p ha k ` q u ´ ` p ha k q ` p ha k ´ q u ` ¨ ¨ ¨ . Notice, however, that if h has odd degree then the action would not arise from the units of the spectrum H Z r u, u ´ s , since the powers of u are weighted by even degrees. On the other hand, for an even degreecohomology class h , the action by hu ´ deg p h q preserves the homogenous degree and gives rise to twist of H ˚ p X ; Z r u, u ´ sq on X . We now characterize the space of twists of periodic integral cohomology, the first summand of which isthe space of twists in the non-periodic case described in [GS17a].
Proposition 1 (Space of twists for periodic integral cohomology) . The space of twists for periodic integralcohomology is B GL p H Z r u, u ´ sq » K p Z { , q ˆ ź k ą K p Z , k ` q . Proof.
We will show that we have an equivalence(2.3) Z { ˆ ź k ą K p Z , k q » GL p H Z r u, u ´ sq . The connected cover of H Z r u, u ´ s is given by H Z r u s and the infinite loop space is the Dold-Kan image of thepositively graded complex Z r u s – ś k Z r k s , which is a model for ś k K p Z , k q . Since the group of units of Z are Z { – t´ , u , we see that GL p H Z r u, u ´ sq is as claimed and delooping gives the desired equivalence. ✷ We now describe the twists via module spectra. For a ring spectrum R , let us recall the Picard -groupoidPic top R from [GS17b], following [BN14]. This is the infinity groupoid whose objects are invertible R -modulespectra. The corresponding geometric realization decomposes in the category of spaces as | Pic top R | » B GL p R q ˆ π Pic top R . For the spectrum H Z r u, u ´ s , Proposition 1 then gives a canonical map K p Z { , q ˆ ś k ą K p Z , k ` q (cid:31) (cid:127) / / | Pic top H Z r u,u ´ s | , given by the inclusion at the identity component of the Picard space. This indeed allows us to twist periodicintegral cohomology by any odd degree integral class ( Z { T p S pace q and the pullbacks of this universal bundle by a map h : X Ñ Pic top R gave a bundle of spectrarepresenting the twisted theory. Since we would like to be as concrete as possible, and relying on as littleabstract machinery as possible, we note that in the present case this universal bundle will take on a relativelysimple form; see the map (2.5).We begin by describing a convenient category in which our bundles of spectra live. Ordinary vectorbundles are allowed to live in topological spaces since the fiber itself is a topological space. However, aspectrum is a generalization of certain types of topological spaces, namely infinite loop spaces. An infiniteloop space is allowed to have homotopy groups in negative degrees and, therefore, cannot itself be regardedas a space. As mentioned above, the convenient category in which our constructions take place is the tangent S pace is the category of compactly generated, weakly Hausdorff spaces. The only reason for this condition on our topologicalspaces is so that we have a convenient category of spaces in which to work. In particular, we have internal mapping spaces inthis category which turns this category into an -category. nfinity category of spaces, T p S pace q (see Remark 2 below) . Similar to the way one defines a spectrum froma prespectrum, we have the following definitions and properties. Definition 2 (Parametrized (pre)spectra) . (i) A parametrized prespectrum over X is a collection of mapsbetween spaces p n : E n Ñ X , n P Z , together with a choice of section, and which come equipped withmorphisms Σ X E n Ñ E n ` , commuting with the sections. Moreover, the following diagram Σ X E n / / Σ X p p n q % % ▲▲▲▲▲▲▲ E n ` p n ` y y sssssss X is required to commute up to a choice of equivalence. The operation Σ X is the result of the pushout E n p n / / p n (cid:15) (cid:15) X (cid:15) (cid:15) X / / Σ X E n . (ii) A parametrized prespectrum t p n : E n Ñ X u for which the adjoint maps E n Ñ Ω X E n ` are equivalencesis called a parametrized spectrum . Note that the above pushout Σ X E n is a generalization of the usual suspension Σ E n , which arises bytaking X to be a point. Remark 2 (The tangent infinity category of spaces from parametrized spectra) . Forgetting about the levelsof the object t p n u , we will often denote a parametrized spectrum whose levels are p n : E n Ñ X , simply by p : E Ñ X . Given two objects p : E Ñ X and p : E Ñ X we have a (ordinary, unparametrized) spectrumof maps Map p p, p q , which in particular (at level zero) gives a collection of homotopy commutative diagrams $’&’% E n / / (cid:15) (cid:15) E n (cid:15) (cid:15) (cid:15) (cid:15) X / / X ,/./- . The resulting structure is that of a stable -category, which we denote by T p S pace q . We denote the subcat-egory on those spectra parametrized over a fixed space X as T p S pace q X . Now that we have a convenient category of parametrized spectra in which to work, we can define a bundleof spectra as follows.
Definition 3 (Bundle of spectra) . A bundle of spectra π : E Ñ X over a space X with fiber a (ring)spectrum R is an object in T p S pace q satisfying the following properties: (i) Fiber:
For each x P X , the pullback π ´ p x q / / (cid:15) (cid:15) E (cid:15) (cid:15) ˚ x / / X is equivalent to R . (ii) Local trivialization:
There is a covering t U α u of X such that, for each U α , we have a Cartesian square(i.e. p8 , q -pullback square) U α ˆ R φ α / / (cid:15) (cid:15) E (cid:15) (cid:15) U α / / X . h α s { ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ The map φ α and the homotopy h α filling the diagram constitute what we call a local trivialization. After some identifications the definition is essentially [MS06, Def. 11.2.3], in the case G “ ˚ . The corresponding -categorytheoretic treatment can be found in [Jo08, Sec. 35.5]. Note that the map φ α and the homotopy h α are not enough to reconstruct the bundle. One also needs to consider pullbacksto higher-fold intersections and higher homotopies filling the resulting diagrams. This is in contrast to the case of an ordinaryvector bundle, where it is enough to know the local trivializations. n the absence of any geometry, bundles of spectra behave more like covering spaces than like smoothvector bundles. The next example illustrates this point. Example 1 (Bundle of spectra over the circle) . Let Z Ñ S be the disconnected cover of S , splitting asthe disjoint union Z “ š k W , where W is the connected cover classified by the subgroup Z Ă Z – π p S q .This cover can be viewed as a Z -subbundle of the M¨obius bundle given by restricting to integers. Viewing S as the unit circle in the complex plane and removing the points ´ and from S , we get correspondingopen sets U and V , respectively, covering S . Over U and V , we have equivalences φ U : Z | U » Z ˆ U and φ V : Z | V » Z ˆ V , which can be chosen so that the transition functions act by multiplication by ´ on the fibers, i.e., φ UV p n, x q “p´ n, x q . The map ´ ˆ : Z Ñ Z extends to a map ´ ˆ : H p Z r u, u ´ sq Ñ H p Z r u, u ´ sq degreewise. Gluingby this automorphism gives the identification $’’&’’% Z (cid:15) (cid:15) S ,//.//- » colim $’’’’&’’’’% H p Z r u, u ´ sq ˆ U X V (cid:15) (cid:15) ´ i / / i / / H p Z r u, u ´ sq ˆ U š H p Z r u, u ´ sq ˆ V (cid:15) (cid:15) U X V / / / / U š V ,////.////- , where i is induced by the usual inclusion into the second factor and the top map ´ i applies the automorphism ´ and then includes into the first factor. This colimit takes place in the category T p S pace q . As part of thedata of the colimit, we have local trivializations φ U : Z | U » H p Z r u, u ´ sq ˆ U and φ V : Z | V » H p Z r u, u ´ sq ˆ V , turning Z into a corresponding bundle of spectra, with fiber H p Z r u, u ´ sq , over S . The transition func-tions take the form φ UV p x, ´q “ ´ , where ´ is the automorphism of the fiber H p Z r u, u ´ sq induced bymultiplication by ´ . In Example 1, the automorphisms φ UV p x, ´q had degree zero, in the sense that they were genuine 1-morphisms and not higher simplices in the space of automorphisms GL p H p Z r u, u ´ qq . The next example,however, gives an instance where we do have higher simplices. Example 2 (Bundle of spectra over the 3-sphere) . Consider the 3-sphere S , equipped with the cover t U, V u obtained by removing the north and south poles, respectively. The intersection U X V » S and, given ouridentification of the units in (the proof of ) Proposition 1, we have π p GL p H Z r u, u ´ sqq » π p K p Z , qq » Z , with generator u . Consequently, the homotopy class of a map U X V » S Ñ GL p H Z r u, u ´ sq is representedby an integer n times the generator u . Via the action of GL p H Z r u, u ´ sq , such a representative gives riseto a map (2.4) nu : S ÝÑ Map ` H Z r u, u ´ s , H Z r u, u ´ s ˘ , and we would like to take the map as supplying the transition data for a bundle on S . Acting by this mapand then by the usual inclusion map U X V ã Ñ U š V into the second factor gives the two top arrows (thatis, ˆ nui and i ), respectively, in the following diagram $’’&’’% Z (cid:15) (cid:15) S ,//.//- » hocolim $’’’’’’’’’’’’’&’’’’’’’’’’’’’% H p Z r u, u ´ sq ˆ U X V (cid:15) (cid:15) ˆ nui ) ) i H p Z r u, u ´ sq ˆ U š H p Z r u, u ´ sq ˆ V (cid:15) (cid:15) U X V / / / / U š V (cid:31) ' y (cid:1) ❴ * ,/////////////./////////////- . he fact that this diagram has nontrivial homotopies filling it, i.e., the ones provided by the map (2.4) , iswhat separates it from Example 1. The homotopy class of sections of the bundle Z Ñ S computes thetwisted cohomology groups. In the same way that ordinary vector bundles with G -structure are classified by maps to the classify-ing space BG , bundles of spectra with fiber R are classified by maps to B GL p R q . There is a universalbundle of spectra over this space. In the present case (i.e. for periodic integral cohomology) it takes thefollowing form. The action of each factor K p Z , k q on the spectrum H Z r u, u ´ s gives rise to a quotient H Z r u, u ´ s{{ K p Z , k q . This leads to the following bundle(2.5) H Z r u, u ´ s{{ K p Z , k q ÝÑ K p Z , k ` q , which we can think of as a universal bundle. Given a map h : X Ñ K p Z , k ` q , we consider the pullbackdiagram(2.6) Z h / / (cid:15) (cid:15) H Z r u, u ´ s{{ K p Z , k q (cid:15) (cid:15) X h / / K p Z , k ` q . Then Z h Ñ X is itself a bundle of spectra with fiber H Z r u, u ´ s . Indeed, the Pasting Lemma for pullbacksimplies that we have a double pullback square(2.7) H Z r u, u ´ s / / (cid:15) (cid:15) Z h / / (cid:15) (cid:15) H Z r u, u ´ s{{ K p Z , k q (cid:15) (cid:15) ˚ / / X h / / K p Z , k ` q , so that H Z r u, u ´ s is identified as the fiber. Next, suppose that X admits a good open cover t U α u . Then, bythe Borsuk Nerve Theorem (see, e.g., [Pr06, Theorem 3.21]), X is homotopy equivalent to the colimit overthe ˇCech nerve of a good open cover t U α u . By iterating pullbacks, we therefore get induced commutativesimplicial diagrams(2.8) ¨ ¨ ¨ / / / / / / š αβ H Z r u, u ´ s ˆ U αβ / / / / (cid:15) (cid:15) š α H Z r u, u ´ s ˆ U α / / (cid:15) (cid:15) Z h / / (cid:15) (cid:15) H Z r u, u ´ s{{ K p Z , k q (cid:15) (cid:15) ¨ ¨ ¨ / / / / / / š αβ U αβ / / / / š α U α / / X h / / K p Z , k ` q , where the bottom simplicial diagram is induced by the ˇCech nerve. Via descent, the top simplicial diagramin (2.8) is homotopy colimiting and this says that (up to homotopy equivalence) we can recover the totalspace Z h by gluing together local trivializations via compatibility maps defined on various intersections. This association gives the following correspondence.
Proposition 4 (Characterization of twisted periodic integral cohomology) . There is a bijective correspon-dence between homotopy classes of maps h : X Ñ K p Z , k ` q and equivalence classes of bundles of spectrawith fiber H Z r u, u ´ s , which admit a K p Z , k q -structure, i.e., a reduction of the structure -group from GL p H Z r u, u ´ sq to K p Z , k q . Example 3 (Classifying map for bundles of spectra over S .) . In Example 2 the transition data specifiedby the map ˆ nu : S » U X V Ñ K p Z , q ã Ñ GL p H Z r u, u ´ sq corresponds to a map h : S Ñ K p Z , q ã Ñ B GL p H Z r u, u ´ sq by the loop-suspension adjunction. This map is the classifying map of the bundle con-structed in that example. Since we are in an p8 , q -category, quotients are taken in the p8 , q -sense, i.e., up to coherence homotopy. The tangent -category of spaces is an example of an -topos and such infinity categories are characterized axiomatically viathe Giraud-Rezk-Lurie axioms [Lu09, Sec. 6.1.5]. One of these axioms is that of descent, which asserts that whenever we havea diagram of the above form with the bottom simplicial diagram being colimiting, and all squares being Cartesian, then thetop simplicial diagram is also colimiting. s we stated in Remark 2, the sections of the map p : Z h Ñ X form a spectrum. Given that, locally, Z h trivializes as H Z r u, u ´ s ˆ U α when t U α u is a good open cover of a space X , we can calculate the spectrumvia the local data as the limit of spectra(2.9) Γ p X ; Z h q “ lim ! ¨ ¨ ¨ š αβ H Z r u, u ´ s o o o o o o š α H Z r u, u ´ s o o o o ) , where again the simplicial homotopy commutative diagram is determined by the transition functions andhigher transition data. In practice, this can aid in calculation; however, it is more useful to develop some ofthe basic properties of the spectrum of sections. Indeed, we will do this in Section 2.2.We finish our current discussion by defining the underlying twisted cohomology groups for twisted periodic Z -cohomology. Notice that, since the fibers H Z r u, u ´ s are 2-periodic, in the sense that Σ H Z r u, u ´ s » H Z r u, u ´ s , and the action by K p Z , k q commutes with this shift, the sections of Z h are also 2-periodic.This leads us to the following definition for the reduced cohomology. Definition 5 (Twisted periodic integral cohomology) . Let h : X Ñ K p Z , k ` q be be a twist for periodicintegral cohomology. We define the h -twisted integral cohomology as the Z { -graded group r H ˚ p X ; h q : “ π ˚ Γ p X, Z h q . We will refer to the degree of a class as either even or odd , corresponding to the identity and nonidentityelements in Z { , respectively. Properties of twisted periodic integral cohomology.
In this section, we state some of the basicproperties of twisted periodic cohomology, which we generalize to twisted periodic smooth Deligne cohomol-ogy in Section 4.2. The following proposition holds more generally for any twisted cohomology theory, butwe will only state this in the present case for the reduced theory r H ˚ p X ; h q . Proposition 6 (Properties of twisted periodic integral cohomology) . Let X be a space and fix a twist as amap h : X Ñ K p Z , k ` q . Consider the category of such pairs p X, h q , with morphisms f : p X, h q Ñ p
Y, ℓ q given by maps f : X Ñ Y such that r f ˚ ℓ s “ r h s . The assignment p X, h q ÞÑ r H ˚ p X ; h q satisfies the followingproperties: (i) r H ˚ p M ; h q is functorial with respect to the maps f : p X, h q Ñ p
Y, ℓ q . (ii) The functor r H ˚ p´ ; h q satisfies the Eilenberg-Steenrod axioms for a reduced generalized cohomology co-homology theory (i.e., modulo the dimension axiom). In particular, we have a Mayer-Vietoris sequence r H ev p M ; h q / / r H ev p U ; h q ‘ r H ev p V ; h q / / r H ev p U X V ; h q B (cid:15) (cid:15) r H odd p U X V ; h q B O O r H odd p U ; h q ‘ r H odd p V ; h q o o r H odd p M ; h q o o where B is the connecting homomorphism, and the sequence is exact at each entry. (iii) For h : X Ñ K p Z , k ` q a trivial twist (i.e. h » ˚ ) we have an isomorphism r H ˚ p X ; h q – r H ˚ p X ; Z r u, u ´ sq . Proof. (i)
Given a map f : X Ñ Y satisfying the desired compatibility, we have an induced double pullbackdiagram(2.10) Z f ˚ ℓ / / (cid:15) (cid:15) Z ℓ / / (cid:15) (cid:15) H Z r u, u ´ s{{ K p Z , k q (cid:15) (cid:15) X f / / Y ℓ / / K p Z , k ` q , which gives an identification Z f ˚ ℓ » Z h . Consequently, we have an induced morphism of sections f ˚ :Γ p X ; Z ℓ q Ñ Γ p Y ; Z f ˚ ℓ q » Ñ Γ p Y, Z h q . Passing to homotopy groups yields a map f ˚ : r H ˚ p Y ; ℓ q Ñ r H ˚ p X ; h q .It is clear that this assignment takes compositions of morphisms of pairs to compositions of group homo-morphisms. (ii) We now verify the generalized Eilenberg-Steenrod axioms. omotopy invariance . It follows from the universal property of the pullback that when two maps f : p X, h q Ñp
Y, ℓ q and g : p X, h q Ñ p
Y, ℓ q are homotopic, the induced map on sections f ˚ : Γ p X ; Z ℓ q Ñ Γ p Y ; Z f ˚ ℓ q » Ñ Γ p Y, Z h q and g ˚ : Γ p X ; Z ℓ q Ñ Γ p Y ; Z g ˚ ℓ q » Ñ Γ p Y, Z h q are homotopic and, therefore, induce isomorphic mapsat the level of cohomology. Additivity . Let X “ š α X α , then a map h : X Ñ K p k ` q is equivalently a collection of maps h α : X α Ñ K p Z , k ` q . Hence, the spectrum of sections Γ p X, Z h q splits as a product ś α Γ p X α , Z h α q . Since takinghomotopy groups commutes with products, we have an isomorphism r H ˚ p X, h q – ź α r H ˚ p X α , h α q . Exactness . Let i : p A, i ˚ h q ã Ñ p
X, h q be an inclusion and consider the cofiber sequence p A, i ˚ h q ã Ñ p
X, h q Ñ cone p i, ˜ h q . The functor Γ p´ ; Z q sends this homotopy cofiber sequences to homotopy fiber sequences and,therefore, we have a fiber sequence Γ p cone p i q ; Z ˜ h q Ñ Γ p X ; Z h q Ñ Γ p A ; Z i ˚ h q . The associated long exactsequence for the cofiber sequence p U X V, r ˚ UV h q (cid:31) (cid:127) / / p U, r ˚ U h q _ K p Z , k ` q p V, r ˚ V h q / / p U Y V, h q , with r W the restriction to the appropriate open set W , gives the Mayer-Vietoris sequence. (iii) Finally, if the twist h : X Ñ K p Z , k ` q is trivial then, up to homotopy, h factors through the pointinclusion ˚ ã Ñ K p Z , k ` q induced by the zero map 0 Ñ Z . Fixing such a homotopy then taking iteratedpullbacks and using the Pasting Lemma gives pullback squares(2.11) Z h » H Z r u, u ´ s ˆ X / / (cid:15) (cid:15) H Z r u, u ´ s / / (cid:15) (cid:15) H Z r u, u ´ s{{ K p Z , k q (cid:15) (cid:15) X h / / ˚ / / K p Z , k ` q . The bundle equivalence Z h » H Z r u, u ´ s ˆ X then induces an equivalence at the level of global sections,and hence an isomorphism at the level of corresponding reduced theories. ✷ Remark 3 (Reduced vs. unreduced) . Note that we can pass from the reduced theory r H ˚ p X ; h q to theunreduced theory of a triple p X, A, h q with A Ă X as usual, by setting H ˚ p X, A, h q – r H ˚ p X { A ; h q ‘ Z r u, u ´ s . Periodic smooth Deligne cohomology
In this section, we introduce the notion of periodic Deligne cohomology. This will set the stage for thenext section, where we identify the twists of this theory.3.1.
Construction as a cohomology theory.
Just as the Deligne complex is indexed by an integer n P Z ,here we have complexes indexed by elements in Z {
2, which we will call either even or odd, depending on theparity. We let Z denote the locally constant sheaf of Z -valued functions. Definition 7 (Even and odd Deligne complexes) . For ev , odd P Z { , corresponding to the identity andnonidentity components, respectively, we have the two complexes D p ev q : “ ` . . . ÝÑ Z ‘ ź k Ω k ` ÝÑ ź k Ω k ÝÑ Z ‘ ź k Ω k ` looooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooon deg ě ÝÑ ÝÑ Z ÝÑ . . . loooooooomoooooooon deg ă ˘ and D p odd q : “ ` . . . ÝÑ ź k Ω k ÝÑ Z ‘ ź k Ω k ` ÝÑ ź k Ω k loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon deg ě ÝÑ ÝÑ Z ÝÑ . . . loooooooomoooooooon deg ă ˘ , Note that homotopy is a relation between morphisms of pairs p X, h q and must respect the maps to the space of twists. here the Z ’s sit in even degrees in the first complex and in odd degrees in the second. In positive degrees,the differential in both complexes is the usual exterior derivative term-wise and on the copies of Z it is givenby the inclusion map Z ã Ñ Ω ã Ñ ś k Ω k . In negative degrees the differential is trivial. The complexes D p ev q and D p odd q are sheaves of chain complexes on the category of all smooth manifolds,topologized as a site via good open covers. Alternatively, both complexes can be regarded as sheaves of chaincomplexes on any fixed manifold M simply by evaluating on the open subsets of M . This is the familiarsetting in which ordinary smooth Deligne cohomology takes place (e.g. [Br93]). We have the followingnatural definition. Definition 8 (Periodic Deligne cohomology) . We define Z { of a smooth manifold M as the sheaf hypercohomology groups p H ev p M ; Z r u, u ´ sq : “ H p M ; D p ev qq and p H odd p M ; Z r u, u ´ sq : “ H p M ; D p odd qq . The following shows that periodic Deligne cohomology can be calculated easily from the ordinary Delignecohomology groups of a manifold.
Proposition 9 (Calculating periodic Deligne cohomology groups) . Let M be a smooth manifold. There arenatural isomorphisms p H ev p M ; Z r u, u ´ sq – à k p H k p M ; Z q and p H odd p M ; Z r u, u ´ sq – à k p H k ` p M ; Z q . Proof.
We will prove the claim for p H ev p M ; Z r u, u ´ sq . The case for p H odd p M ; Z r u, u ´ sq is proved similarly.To this end, we organize the sheaf of chain complexes D p ev q as follows Z (cid:16) p ! ! ❇❇❇ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ . . . d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ●●●●● . . . Z (cid:16) p ! ! ❇❇❇ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ . . . d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ! ! ❈❈❈ Ω d ●●●●● . . . Z ! ! ❈❈❈❈ Ω " " ❉❉❉❉ Ω " " ❉❉❉❉ Ω " " ❉❉❉❉ . . . ´ " " ❉❉❉❉ " " ❉❉❉❉ " " ❉❉❉❉ $ $ ❍❍❍❍❍❍ . . . ´ Z . . . where the numbers on the vertical axis index the degree of the complex. The diagonal arrows represent thedifferential on each component of the product taken over a given row. The diagonal complexes are easilyseen to be the usual Deligne complex and, therefore, we have a splitting(3.1) D p ev q – ź k D p k q ‘ ź k Z r´ k s , where the second summand comes from the negative degrees of the complex. The latter do not contribute tothe hypercohomology of the complex, as the ˇCech resolution of the complex necessarily vanishes in negativedegrees. Thus, the hypercohomology groups split as desired. ✷ From Proposition 9, it follows immediately that the periodic Deligne cohomology groups fit into a differen-tial cohomology diamond diagram and into exact sequences similar to those for ordinary Deligne cohomology,as an instance of differential integral cohomology [SS08]. The cohomological degrees on both right hand sides is 0, due to the shift in the complexes in Definition 7, in analogy to theusual Deligne case, i.e., expression (1.1). roposition 10 (Periodic Deligne cohomology diamond) . We have the exact diamond diagram (3.2) Ω odd p M q{ im p d q a ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ d / / Ω evcl p M q & & ▼▼▼▼▼▼▼▼▼▼ H odd p M ; R r u, u ´ sq ❧❧❧❧❧❧❧❧❧❧❧❧❧ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ p H ev p M ; Z r u, u ´ sq I & & ▼▼▼▼▼▼▼▼▼▼▼ R qqqqqqqqqq H ev p M ; R r u, u ´ sq H odd p M ; R r u, u ´ s{ Z r u, u ´ sq ♠♠♠♠♠♠♠♠♠♠♠♠♠ β / / H ev p M ; Z r u, u ´ sq j qqqqqqqqqqq for the even Deligne complex and a similar diamond for the odd one, given by switching ev and odd . Here Ω odd p M q and Ω ev p M q are the groups of differential forms of odd and even degrees, respectively. For example,an element ω P Ω ev p M q is a formal combination ω “ ω ` ω ` ω ` ¨ ¨ ¨ , with ω i a differential form of degree i . Remark 4 (Extension of the diamond to a long exact sequence) . One of the diagonals in the diamonddiagram in Proposition 10 can be extended to a long exact sequence. Depending on the parity, the relevantsegments of this long exact sequence are given by H ev p M ; Z r u, u ´ sq ÝÑ Ω ev p M q{ im p d q ÝÑ p H odd p M ; Z r u, u ´ sq ÝÑ H odd p M ; Z r u, u ´ sq ÝÑ ,H odd p M ; Z r u, u ´ sq ÝÑ Ω odd p M q{ im p d q ÝÑ p H ev p M ; Z r u, u ´ sq ÝÑ H ev p M ; Z r u, u ´ sq ÝÑ . The map into the quotient Ω ev p M q{ im p d q takes a periodic integral class and maps it to the class of itscorresponding de Rham representative (i.e. a form with integral periods). Note also that the map R : p H ev p M ; Z r u, u ´ sq Ñ Ω evcl p M q is not surjective; its image is the subgroup of closed forms with integralperiods. Ring structure and examples.
Eventually, we would like to consider the twists of this theory and,to do this, we need a ring structure on this periodic Deligne complex. Recall that for ordinary Delignecohomology, the Deligne-Beilinson cup product gives a collection of morphisms of sheaves of chain complexes[De71][Be85] (see also [FSS13][FSS15a])(3.3) Y DB : D p n q b D p m q ÝÑ D p n ` m q . At the level of local sections, it is defined by the formula α Y DB β “ $&% αβ, deg p α q “ nα ^ dβ, deg p β q “ , deg p α q ‰ n , otherwise . Since the even periodic Deligne complex split as the product (3.1) (and similarly for the odd), there aremultiplication maps(3.4) Y DB : $&% D p ev q b D p ev q ÝÑ D p ev q , D p ev q b D p odd q ÝÑ D p odd q , D p odd q b D p odd q ÝÑ D p ev q , induced by the cup product Y DB from (3.3) in positive degrees and the multiplication of integers in negativedegrees. It is immediate that these maps descend to a graded commutative cup product which is compatiblewith the Deligne-Beilinson cup product term-wise. We summarize these observations as follows. Proposition 11 (Superalgebra structure on periodic Deligne cohomology) . With the multiplication maps (3.4) induced by the Deligne-Beilinson cup product, the complex D p ev q ‘ D p odd q admits the structure of asheaf of differentially graded superalgebras. At the level of hypercohomology, it gives p H ev { odd p M ; Z r u, u ´ sq : “ p H ev p M ; Z r u, u ´ sq ‘ p H odd p M ; Z r u, u ´ sq he structure of a commutative superalgebra. Moreover, we have commutative diagrams p H ev { odd p M ; Z r u, u ´ sq b p H ev { odd p M ; Z r u, u ´ sq Y DB / / (cid:15) (cid:15) p H ev { odd p M ; Z r u, u ´ sq (cid:15) (cid:15) H ev { odd p M ; Z r u, u ´ sq b H ev { odd p M ; Z r u, u ´ sq Y / / H ev { odd p M ; Z r u, u ´ sq and p H ev { odd p M ; Z r u, u ´ sq b p H ev { odd p M ; Z r u, u ´ sq Y DB / / (cid:15) (cid:15) p H ev { odd p M ; Z r u, u ´ sq (cid:15) (cid:15) Ω ev { oddcl p M q b Ω ev { oddcl p M q ^ / / Ω ev { oddcl p M q , where H ev { odd p M ; Z r u, u ´ sq is periodic integral cohomology, endowed with the superalgebra structure inher-ited from the cup product, and Ω ev { odd p M q is the superalgebra of graded differential forms. We now illustrate this with the case of spheres.
Example 4 (Periodic Deligne cohomology of even spheres) . Let S k be the smooth even-dimensional sphere.The underlying periodic integral cohomology is readily computed as H ev p S k ; Z r u, u ´ sq – Z ‘ Z and H odd p S k ; Z r u, u ´ sq – . Given the two long exact sequences in Remark 4, we easily compute p H ev p S k ; Z r u, u ´ sq – Ω odd p S k q{ im p d q ‘ Z ‘ Z and p H odd p S k ; Z r u, u ´ sq – Ω ev p S k q{ Ω evcl , Z p S k q , where Ω evcl , Z p S k q is the subgroup of closed even forms with integral periods (i.e., each component of an element ω “ ω ` ω ` ¨ ¨ ¨ has integral periods). Example 5 (Periodic Deligne cohomology of odd spheres) . Similarly, we calculate for odd spheres using thesame two sequences above, to get in this case p H odd p S k ` ; Z r u, u ´ sq – Ω ev p S k ` q{ im p d q ‘ Z and p H ev p S k ` ; Z r u, u ´ sq – Ω odd p S k ` q{ Ω oddcl , Z p S k ` q ‘ Z , where one of the Z factors has moved, in comparison to the case of even spheres, due to parity reasons. Twisted periodic smooth Deligne cohomology
In this section we turn to twisting periodic Deligne cohomology constructed above. Just as twisted periodicintegral cohomology in Section 2.1 takes the form of a bundle of spectra over a parametrizing space, here wewill have a smooth bundle of spectra, parametrized over a smooth manifold M . In the smooth setting, ourstarting point is no longer the category of spaces and its tangent infinity category T p S pace q , but rather thecategory of smooth stacks S h p M f q and its tangent infinity category T p S h p M f qq .4.1. The parametrized spectrum and gerbes via smooth stacks.
Let M be a smooth manifold andconsider the site of open subsets O pen p M q , topologized via the good open covers t U α Ñ M u . Smooth stackson M are similar to smooth sheaves, but instead of assigning a set of elements to an object U P O pen p M q , weassign a space (usually modeled combinatorially by a simplicial set). The sheaf gluing condition is replacedby a weaker condition, where we only require gluing up to equivalence. To ease the transition, we start withthe following. Example 6 (Gerbe with U p q -band) . Consider a local homeomorphisms U ã Ñ M , with U an open subset of M . To each such map, we assign the groupoid of line bundles with connection. This defines a Dixmier-Douadysheaf of groupoids G on M . This gerbe has U p q -band, and a connective structure on this gerbe is given by achoice of Ω -torsor (satisfying some properties). Using the affine structure on the space of connections, wesee that the sheaf of connections defines such a torsor, so that G admits a connective structure. A curving ofthis structure is an assignment to each line bundle with connection p L, ∇ q , a two-form K p ∇ q to be thoughtof as the curvature. The passage from Brylinski’s gerbe [Br93] to higher smooth stacks is essentially obtained imply by taking the nerve of the sheaf of groupoids G , yielding an -groupoid (which is a combinatorialmodel for a space), although one needs to be careful in keeping track of the connective data and curving (seeExample 7 below). There is a large -category of smooth stacks S h p M f q which does not depend on a choice of underlyingsmooth manifold. The site for this -category is the site of all smooth manifolds M f, topologized via goodopen covers. Any object in S h p M f q can be restricted to a single manifold by simply considering its value onopen subsets U ã Ñ M . One of the benefits of working in this larger -category is that one can define modulistacks X which represent objects of interest over M via maps M Ñ X . For example, the moduli stack ofhigher gerbes with connection B n U p q ∇ was studied in [FSSt12][SSS12][Sc13][FSS15a][FSS15b]. One way topresent this stack is by applying the Dold-Kan functor to the sheaf of chain complexes B n U p q ∇ “ DK ` . . . / / / / U p q d log / / Ω d / / Ω d / / . . . / / Ω n ˘ , where the sheaf U p q : “ C p´ ; U p qq sits in degree n . The sheaf in the argument of DK is quasi-isomorphic(via the exponential map) to the smooth Deligne complex D p n q . The Dold-Kan functor sends quasi-isomorphisms to weak equivalences and (since we are working up to equivalence) this justifies the uniformnotation B n U p q ∇ for both of the resulting stacks (i.e. upon applying DK to either complex). The stack B n U p q ∇ sits in a Cartesian square(4.1) B n U p q ∇ / / (cid:15) (cid:15) Ω n ` (cid:15) (cid:15) B n ` Z / / B k ` R » Ω ď n ` , where Ω ď n ` is the stack presented by the sheaf of chain complexes(4.2) ` . . . / / / / Ω / / Ω d / / Ω d / / . . . / / Ω n ` ˘ . In [FSSt12], it was shown that the homotopy classes of maps M Ñ B n U p q ∇ is in bijective correspondencewith the Deligne cohomology group p H n p M ; Z q . Example 7 (Stack of 2-bundles with connections/gerbes with connections) . The smooth stack B U p q ∇ can be presented via the Dold-Kan correspondence by the sheaf of chain complexes B U p q ∇ “ L ˝ DK ` U p q d log ÝÝÝÑ Ω d ÝÑ Ω ˘ , where L is the stackification functor. Let φ : R n Ñ M be a local chart. For a convex open subset U Ă R n ,this stack can be evaluated on the corresponding open subset V “ φ p U q via Map p V, B U p q ∇ q » DK ` C p V, U p qq d log ÝÝÝÑ Ω p V q d ÝÑ Ω p V q ˘ . More generally, descent for the stack B U p q ∇ implies that, for any choice of good open cover t U α u of M ,the space of maps Map p M, B U p q ∇ q can be identified by replacing M with the ˇCech nerve ˇ C pt U α uq of t U α u and considering instead the space of maps Map ´ ˇ C pt U α uq , DK ` U p q d log ÝÝÝÑ Ω d ÝÑ Ω ˘¯ . By the basic properties of the Dold-Kan correspondence we have an isomorphism π ´ Map ´ ˇ C pt U α uq , DK ` U p q d log ÝÝÝÑ Ω d ÝÑ Ω ˘¯¯ – H p M ; U p q d log ÝÝÝÑ Ω d ÝÑ Ω ˘ . By [Br93, Theorem 5.3.11] , the elements on the right parametrize the homotopy classes of the gerbes withconnective structure and curving considered in Example 6. This is a functor which turns a prestack into a stack, analogously to the way a sheafification functor turns a presheaf into asheaf. See [Lu09, Sec. 6.5.3] for details. The shift in degree occurs because on the left we consider the complex U p q d log ÝÝÝÑ Ω d ÝÝÑ Ω as being shifted up two degreesrelative to the complex appearing on the right. he definition of parametrized spectra in the smooth setting is a direct extension of Definition 2 fromspaces to stacks. Definition 12 (Smooth parametrized spectrum) . A smooth parametrized prespectrum is a collection ofmorphisms p n : E n Ñ M between smooth stacks in S h p M f q , n P Z , with a choice of section, equippedwith morphisms Σ M E n Ñ E n ` , making similar diagrams as in Definition 2 commute up to a choice ofequivalence in S h p M f q . A smooth parametrized prespectrum t p n : E n Ñ M u for which the adjoint maps E n Ñ Ω B E n ` are equivalences is called smooth parametrized spectrum . Remark 5 (Identifying the proper category as a setting) . (i)
Note that Definition 12 is almost verbatim thesame as Definition 2, the only difference being where the objects E n and M live (i.e. smooth stacks insteadof spaces). In this context we still have a mapping spectrum between two smooth spectra. The resultingstructure is again a stable infinity category and we denote this category by T p S h p M f qq . (ii) We will be most concerned with the case when M is a smooth manifold. By the Yoneda embedding, everysmooth manifold embeds as an object in S h p M f q via its sheaf of smooth plots, i.e., the sheaf sending M tothe set of smooth maps N Ñ M , with N any other manifold. (iii) One might wonder why the seemingly complicated -category T p S h p M f qq is necessary to work in. Inparticular, one might think that working with the more familiar category of sheaves of chain complexes shouldbe more transparent. Note, however, that we are naturally led to the - topos T p S h p M f qq for two reasons.First, passing to sheaves is necessary to capture the geometry of the de Rham complex (which is a crucialingredient in defining Deligne cohomology). Second, in contrast to the category of sheaves of chain complexes,the axioms of an -topos (in particular descent) make it a convenient setting to talk about bundles. We have the following natural definition for a smooth, locally trivial bundle of spectra.
Definition 13 (Smooth bundle of spectra) . Let M be a smooth manifold. A smooth bundle of spectra π : E Ñ M over M with fiber the sheaf of spectra R is an object in T p S h p M f qq M satisfying the sameproperties as in Definition 3 with M replacing X . We now wish to focus our scope to the case of periodic Deligne cohomology. Consider the sheaf of ringspectra given by applying the Eilenberg-MacLane functor H to the sheaf of chain complexes D p ev q and D p odd q . In Section 3.1, we saw that this ring spectrum represents periodic Deligne cohomology, in the sensethat p H ev p M ; Z r u, u ´ sq – π Map ` M ; H p D p ev qq ˘ and p H odd p M ; Z r u, u ´ sq – π Map ` M ; H p D p odd qq ˘ . We would like to identify a large class of twists for this theory. To this end, let us consider the stack of twistsin diagram (1.4) with p R the periodic differential ring spectrum given by both H p D p ev qq and H p D p odd qq ,separately. At first, it might appear that we would get two stacks of twists corresponding to both the evenand odd degrees; however, this is not the case. Proposition 14 (Equivalence of stacks of even and odd twists for periodic Deligne cohomology) . We havea canonical equivalence of smooth stacks y Tw H p D p odd qq » y Tw H p D p ev qq , induced by shifting both the ring spectrum H p Z r u, u ´ sq and the invertible periodic de Rham complex Ω ˚ r u, u ´ s up by one degree each. Proof.
It is clear formally that shifting a module spectrum R τ up by one degree is a module spectrum overthe ring spectrum R , i.e., the module maps µ : R m ^ R nh Ñ R m ` nh give rise to maps R m ^ R n ` h Ñ R m ` n ` h .Similarly, shifting a K-flat invertible module L is again a K-flat invertible module. Moreover, given anyequivalence H p L q » R τ ^ H R , we get a corresponding equivalence at the level of the shifts. By the universal property of the pullback, wehave an induced map at the level of the twists. For smooth periodic Deligne cohomology this takes the form y Tw H p D p odd qq ÝÑ y Tw H p D p ev qq . Note that the mapping spectra are not smooth or parametrized; they are ordinary topological spectra. t is immediate that this map admits an inverse induced by shifting down. ✷ Proposition 14 implies that we do not have to consider the even and odd degrees separately, but we canview a given twist as corresponding to either spectrum. Henceforth, we will only refer to the stack of twistsof periodic Deligne cohomology and denote the stack simply by y Tw.
Remark 6 (Chern-Simons type hierarchy of trivilizations of ˇCech-Deligne cocycles) . Consider a ˇCech-Deligne cocycle η “ p η p q , η p q , η p q , ¨ ¨ ¨ , η p k ` q q on a smooth manifold M , where η p i q is the cocycle dataon the i -fold intersection, i.e., η p q is a k -form defined on open sets, η p q is a p k ´ q form definedon intersections, etc. To this we associate automorphisms and higher automorphisms of the periodic deRham complex Ω ˚ r u, u ´ s on a smooth manifold M . More precisely, we associate to such a cocycle theautomorphisms ch p q p η q ^ p´q “ e η p q ^ p´q “ ` η p q ^ p´q ` η p q ^ η p q ^ p´q ` η p q ^ η p q ^ η p q ^ p´q ` ¨ ¨ ¨ , CS p q p η q ^ p´q “ η p q ` η p q ^ dη p q ^ p´q ` η p q ^ dη p q ^ dη p q ^ p´q ` ¨ ¨ ¨ , ...where we have a primary invariant ch p q p η q , then a secondary invariant CS p q p η q for the latter, then a tertiaryinvariant for the latter, and a similar pattern in higher degrees, obtained by using the various cocycle data for η . This assignment is explained in the proof of [GS17b, Theorem 17] and in the discussion leading up to thattheorem. In particular, taking k “ , we get η “ p B α , A αβ , f αβγ , n αβγ,δ q , which is the ˇCech-Deligne cocyclecorresponding to the ‘standard’ gerbe with connection (see Example 6), encoding a twist. These represent thehomotopies, homotopies between homotopies, etc., respectively, in diagram (4.4) of Example 8 below. Theexpression for CS p q p η q is a sum of higher product abelian Chern-Simons theories, in the sense of [FSS13] .The next terms (not explicitly recorded for brevity) correspond to tertiary and higher structures, in the senseof [FSS13][Sa14] . Proposition 15 (Twisting periodic Deligne cohomology by odd degree gerbes with connection) . Let y Tw p M q denote the stack of twists, evaluated on a smooth manifold M . Then every ˇCech-Deligne cocycle of degree k ` defines a twist of periodic Deligne cohomology. In fact, there is a morphism of smooth stacks B k U p q ∇ ÝÑ y Tw , refining the map K p Z , k ` q Ñ B GL p H Z r u, u ´ sq ã Ñ Pic top H Z r u,u ´ s . Proof.
The stack B k U p q ∇ fits into the Cartesian square (4.1). This pullback in smooth stacks can becomputed by the stackification of the corresponding pullback in prestacks, which is computed objectwise.Moreover, a morphism of prestacks into a stack is, equivalently, a morphism of stacks out of the stackifi-cation. It, therefore, suffices to construct the map objectwise out of the three stacks Ω k ` , B k ` Z andΩ ď k ` and for every (1-)homotopy filling the diagram, a corresponding homotopy filling diagram (1.4).To that end, fix an arbitrary manifold M and define the three mapsΩ k ` p M q / / Pic form p M q B k ` R p M q / / B GL p H R r u, u ´ sqp M q (cid:31) (cid:127) / / Pic dR p M q B k ` Z p M q / / B GL p H Z r u, u ´ sqp M q (cid:31) (cid:127) / / Pic top p M q as follows. The first map sends a closed odd-degree form to the invertible module over the periodic de Rhamcomplex, p Ω ˚ r u, u ´ sp M q , d H q , where the differential d H “ d ` H ^ acts on a differential form as(4.3) d H p ω q “ d H p ω ` ω ` ¨ ¨ ¨ q “ dω ` dω ` ¨ ¨ ¨ ` p H ^ ω ` dω k q ` ¨ ¨ ¨ . The second map is induced by the canonical inclusion map K p k ` , R q ã Ñ B GL p H R r u, u ´ sq , while thethird map is similarly induced by K p k ` , Z q ã Ñ B GL p H Z r u, u ´ sq .A homotopy filling the diagram (4.1) can be identified via the Dold-Kan correspondence as an element η in degree 1 of the total complex of the ˇCech double complex Tot ` t U α u , Ω ď k ` ˘ , with t U α u a good open This follows from the adjunction i $ L , with L the stackification functor and i the inclusion functor. over of M , subject to the following condition: We require that D p η q “ H ´ h , with H the globally definedclosed differential form of degree 2 k ` k ` p M q and h the real-valued ˇCech cocycle ofdegree 2 k ` B k ` Z p M q ã Ñ B k ` R p M q (see [FSSt12] for more details on thelatter stack). To the homotopy η , we need to construct a corresponding equivalence H p Ω ˚ r u, u ´ s , d H q » H R r u, u ´ s h , where H R r u, u ´ s h is the locally constant sheaf of spectra corresponding to an invertible module over H R r u, u ´ s . The former is classified by the map h : M / / K p Z , k ` q / / K p R , k ` q (cid:31) (cid:127) / / Pic dR H R r u,u ´ s . As described in [GS17b], the locally defined form ch p q p η q from Remark 6 defines a local trivializationch p q p η q : p Ω ˚ r u, u ´ s , d H q| U α » ÝÑ Ω ˚ r u, u ´ s| U α . The higher CS p i q ’s in Remark 6 correspond to automorphisms on intersections and higher automorphismson higher intersections. As in the proof of [GS17b, Theorem 17], this cocycle determines an edge in Pic dR connecting H p Ω ˚ r u, u ´ s , d H q and H R r u, u ´ s h . It is clear from the construction that this map refines theinclusion of K p Z , k ` q into the topological twists. ✷ In Section 2, we described a universal bundle of spectra which classifies bundles of spectra over a space X with a K p Z , k q -structure prescribed by a twist h : X Ñ K p Z , k ` q . There is a similar universal bundleover the stack of twists (1.4), described in [GS17b], which classifies twisted differential cohomology theoriesvia pullback. Descent allows us to glue together the bundle via local trivializations. As a fundamentalexample, we consider the following. Example 8 (Higher-twisted differential forms as sections of a smooth bundle of spectra) . Consider the sheafof complexes p Ω ˚ r u, u ´ s , d H q on a smooth manifold M , which is degreewise identical to the periodic complexof forms, but which is equipped with the differential d H : “ d ` H ^ , acting by (4.3) . Here, H a closed formof degree k ` . Applying the Eilenberg-MacLane functor H to p Ω ˚ r u, u ´ s , d H q gives a sheaf of spectraon M . Now p Ω ˚ r u, u ´ s , d H q is an invertible module over p Ω ˚ r u, u ´ s , d q which is locally equivalent (by thePoincar´e Lemma) to the constant sheaf R r u, u ´ s . Thus H p Ω ˚ r u, u ´ s , d H q gives a sheaf of spectra which isa module over H R r u, u ´ s . Pulling back the universal bundle of spectra λ ÝÑ Pic dR H R r u,u ´ s (see [GS17b] for this construction) by the map τ : M Ñ Pic dR H R r u,u ´ s , which picks out the twisted sheaf ofspectra H p Ω ˚ r u, u ´ s , d H q , gives a smooth bundle of spectra E Ñ M . The sheaf of local sections of the latterevaluated on U is, by definition, H p Ω ˚ r u, u ´ s , d H qp U q . Choose local potentials B α for H on each elementof a good open cover t U α u of M (i.e. dB α “ H ). Then, on each patch U α , we have quasi-isomorphisms ofsheaves of complexes e B α ^ : p Ω ˚ r u, u ´ s , d H q| U α » ÝÑ Ω ˚ r u, u ´ s| U α which send a local section ω to the wedge product with the formal exponential e B α “ ` B α ` B α ` ¨ ¨ ¨ . These quasi-isomorphisms correspond to local trivializations e B α ^ : E | U α ÝÑ H p Ω ˚ r u, u ´ sq ˆ U α . In fact, a choice of representative of H in the ˇCech-de Rham double complex gives rise to a homotopycommutative diagram (4.4) . . . / / / / / / / / š αβγ H p Ω ˚ r u, u ´ sq ˆ U αβγ / / / / / / š αβ H p Ω ˚ r u, u ´ sq ˆ U αβ / / / / š α H p Ω ˚ r u, u ´ sq ˆ U α where the simplicial maps at each stage are determined by the ˇCech-de Rham data for H . For example,choosing a differential form A αβ on intersections, satisfying dA αβ “ B α ´ B β , gives rise to a homotopy ommutative diagram H p Ω ˚ r u, u ´ sq ˆ U αβ id / / e ´ Bα ) ) ❙❙❙❙❙❙❙❙❙❙❙ H p Ω ˚ r u, u ´ sq ˆ U αβ E | U αβ e Bβ ❦❦❦❦❦❦❦❦❦❦❦ , with homotopy given by wedge product with the abelian Chern-Simons form (cf. Remark 6) CS p A αβ q “ A αβ ` A αβ ^ dA αβ ` A αβ ^ dA αβ ^ dA αβ ` ¨ ¨ ¨ so that dCS p A αβ q “ e dA αβ “ e B α ´ B β “ e B α e ´ B β . By descent, one concludes that E is, in fact, a colimit over this diagram. Notice also that, since the globalsections of E are H p Ω ˚ r u, u ´ sp M q , d H q , we immediately have that the twisted cohomology represented bythe bundle H is the twisted de Rham cohomology of M . Remark 7 (The twisted periodic Deligne complex as sections of a smooth bundle of spectra) . Example 8shows that, secretly, elements in the twisted de Rham complex are sections of a bundle of spectra over M .That example can be adapted to the periodic Deligne complex by noting that if the de Rham twist H hasintegral periods, then the corresponding topological twist h : M Ñ K p R , k ` q factors (up to homotopy)through K p Z , k ` q . In this case, the triple ` H Z r u, u ´ s h , t, p Ω ˚ r u, u ´ s , d H q ˘ , with t : H R r u, u ´ s h » H p Ω ˚ r u, u ´ s , d H q the twisted de Rham equivalence, gives a twist of periodic Deligne cohomology on M , hence a map τ : M Ñ y Tw . Pulling back by the universal bundle ˆ λ Ñ y Tw gives similar data to the ones in Example 8. Definition 16 (Twisted periodic Deligne cohomology) . Let H be a closed differential form of odd degreewhich has integral periods. Then H can be lifted to a map ˆ h : M ÝÑ B k U p q ∇ . According to Proposition 14, we can regard this as either twisting H p D p ev qq or H p D p odd qq . Let Z evˆ h Ñ M and Z oddˆ h Ñ M be the corresponding smooth bundles of spectra. We define the twisted periodic Delignecohomology to be the homotopy classes of sections of the corresponding bundle Z ev { oddˆ h Ñ M , i.e., p H ev p M ; ˆ h q : “ π Γ p M ; Z evˆ h q and p H odd p M ; ˆ h q : “ π Γ p M ; Z oddˆ h q . Just as one can define tensor product and direct sum of vector bundles, one can similarly define thewedge product and smash product of bundles of spectra (see [GS17b] for the definition of the smash product;the wedge product is defined similarly). In the present case, we have a Z { E ev _ E odd Ñ M . The local sections of this bundle are given by evaluating the wedge product of spectra H p D p ev qq _ H p D p odd qq » H p D p ev q ‘ D p odd qq on open subsets U Ă M . Given the multiplicative structureof periodic Deligne cohomology from Section 3.2, we have the following. Proposition 17 (Module structure of twisted periodic Deligne cohomology) . The sheaf of sections of thewedge bundle E ev _ E odd Ñ M is a module spectrum over the sheaf of ring spectra given by the wedge H p D p ev qq _ H p D p odd qq . The module action descends to a map µ : p H ev { odd p M ; Z r u, u ´ sq b p H ev { odd p M ; ˆ h q / / p H ev { odd p M ; ˆ h q , turning p H ev { odd p M ; ˆ h q into a module over the superalgebra p H ev { odd p M ; Z r u, u ´ sq . .2. Properties of twisted periodic smooth Deligne cohomology.
In this section, we list some of theproperties of twisted periodic Deligne cohomology. Most of these properties are similar to those discussed inProposition 6. A main point of contrast to emphasize here is that this theory will not satisfy the homotopyinvariance axiom, as is the case for any differential cohomology theory (see [BS10]).
Proposition 18 (Properties of twisted periodic Deligne cohomology) . Let M be a smooth manifold and fixa twist ˆ h : M Ñ B k U p q ∇ . Consider the category of such pairs p M, ˆ h q , with morphisms f : p M, ˆ h q Ñ p M, ˆ ℓ q given by smooth maps f : M Ñ N such that r f ˚ ˆ ℓ s “ r ˆ h s . The assignment p M, ˆ h q ÞÑ p H ˚ p M ; ˆ h q satisfies thefollowing properties: (i) p H ˚ p M ; ˆ h q is functorial with respect to the maps f : p M, ˆ h q Ñ p N, ˆ ℓ q . (ii) The functor p H ˚ p´ ; ˆ h q satisfies the Eilenberg-Steenrod axioms (modulo the dimension axiom and homo-topy invariance!) for a reduced cohomology theory. In particular, we have a Mayer-Vietoris sequencewhich takes the form ... / / H ˚´ R { Z p U ; h q ‘ H ˚´ R { Z p V ; h q / / H ˚´ R { Z p U X V ; h q EDBCGF@A / / p H ˚ p M ; ˆ h q / / p H ˚ p U ; ˆ h q ‘ p H ˚` p V ; ˆ h q / / p H ˚ p U X V ; ˆ h q EDBCGF@A / / H ˚` p M ; h q / / H ˚` p U ; h q ‘ H ˚` p V ; h q / / ... (iii) For ˆ h : M Ñ B k U p q ∇ a trivial twist (i.e. ˆ h » ˚ in smooth stacks) we have an isomorphism (4.5) p H ˚ p M ; ˆ h q – p H ˚ p M ; Z r u, u ´ sq . Even more strongly, we still have an isomorphism (4.5) if just the underlying topological twist h : M Ñ K p Z , k ` q is trivial. Proof. (i)
Given a map f : M Ñ N satisfying the desired compatibility, we have an induced double pullbackdiagram(4.6) Z f ˚ ˆ h / / (cid:15) (cid:15) Z ˆ h / / (cid:15) (cid:15) ˆ λ (cid:15) (cid:15) M f / / N ˆ h / / B k U p q ∇ / / y Tw , which gives the identification Z f ˚ ˆ h » Z ˆ ℓ . As a consequence, we have an induced morphism of sections f ˚ : Γ p N ; Z ˆ h q Ñ Γ p M ; Z ˆ ℓ q . Passing to homotopy groups yields a map f ˚ : p H ˚ p N ; ˆ ℓ q Ñ p H ˚ p M ; ˆ h q . (ii) We now verify the applicable Eilenberg-Steenrod axioms.
Additivity . Let M “ š α M α with each M α a smooth manifold. A map ˆ h : M Ñ B k U p q ∇ is equivalentlya collection of maps h α : M α Ñ B k U p q ∇ . Then the spectrum of sections Γ p M, Z ˆ h q splits as a product ś α Γ p M α , Z ˆ h α q . Since taking homotopy groups commutes with products, we have an isomorphism p H ˚ p M, ˆ h q – ź α p H ˚ p M α , ˆ h α q . Exactness . This follows verbatim as in the proof of Proposition 6, with the space X replaced by a smoothmanifold M , A Ă M a submanifold, and the map h : X Ñ K p Z , k ` q replaced by the refinementˆ h : M Ñ B k U p q ∇ . (iii) Finally, if the twist ˆ h : M Ñ B k U p q ∇ is topologically trivial, i.e., its geometric realization h : | M | » M Ñ | B k U p q ∇ | » K p Z , k ` q is homotopic to a the constant map induced by 0 Ñ Z . In this case, the nderlying twisted spectrum H Z r u, u ´ s h is equivalent to H Z r u, u ´ s and we have the diagram H p Z r u, u ´ sq » / / ^ H R (cid:15) (cid:15) H p Z r u, u ´ sq h ^ H R (cid:15) (cid:15) H p R r u, u ´ sq » c (cid:15) (cid:15) » / / H p R r u, u ´ sq h » t (cid:15) (cid:15) H p Ω ˚ r u, u ´ sq » / / H p Ω ˚ r u, u ´ s , d H q , where the bottom equivalence depends on a choice of homotopy inverse for c and is defined as the obviouscomposition in the diagram. By the basic properties of the functor H (see [BN14, pp. 17-18] for discussion),the existence of the bottom equivalence implies that Ω ˚ r u, u ´ s and p Ω ˚ r u, u ´ s , d H q are connected by a zig-zag of quasi-isomorphisms. This is manifestly the data needed to define an equivalence in the -groupoid y Tw p M q . ✷ The spectral sequence { AHSS h and examples In this section, we apply the twisted Atiyah-Hirzebruch spectral sequence (both the classical [Ro82][Ro89][AS06] and the differential refinement [GS17b]) to calculate the twisted periodic integral and Deligne coho-mology of spheres.5.1.
The spectral sequence in twisted periodic smooth Deligne cohomology.
In [AS06], the firstnonvanishing differential for the twisted AHSS (applied to K -theory) on a space X was identified by observingthat the only degree three increasing operations for spaces equipped with maps X Ñ K p Z , q are given bythe cohomology group H n ` p K p Z , n q ˆ K p Z , qq – H n ` p K p Z , n qq ‘ H n ` p K p Z , qq ‘ Z . The third factor on the right hand side is generated by the product of the generators for H n p K p Z , n qq and H p K p Z , qq . From this, one deduces that d p x q “ Sq Z p x q ´ r h s Y x , with r h s the twisting integral class and Sq Z the third integral Steenrod square, which comes from theuntwisted AHSS for K -theory [AH62].The situation for periodic integral cohomology is much easier since the untwisted differentials vanish andsince one is able to compare easily with rational cohomology. Considering again a degree three twist h , wethen find that d p x q “ ´r h s Y x . The same argument applies not only in the degree three case, but also in higher odd degrees. This is due tothe fact that for spaces equipped with maps X Ñ K p k ` Z q , we again have the identification H n ` k ` p K p Z , n q ˆ K p Z , k ` qq – H n ` k ` p K p Z , n qq ‘ H n ` k ` p K p Z , k ` qq ‘ Z , with the last factor being generated by the product of the generator of H n p K p Z , n qq and the generator of H k ` p K p Z , k ` qq . We, therefore, have the following. Proposition 19 (First differential for AHSS h for twisted periodic integral cohomology) . Let h : X Ñ K p Z , k ` q be a twist of periodic integral cohomology. Then the first nonvanishing differential in theassociated AHSS occurs on the E k ` -page and is given by d k ` p x q “ ´r h s Y x . We will illustrate this in Examples 9 and 10 below.In [GS17b], we developed an AHSS for twisted differential cohomology theories, in turn generalizing thatof a differential theory [GS16b]. In the case of periodic Deligne cohomology, there are two spectral sequencescorresponding to the even and odd degrees (separately). This can be deduced, for example, from the K -theory differential and the fact that (on spheres) the Chern character lands inintegral cohomology. emma 20 (The E -page for even degrees in { AHSS ˆ h for twisted periodic Deligne cohomology) . The E -pagefor the even case looks as follows (5.1) 10 Ω ev d H - cl , Z p M q´ H p M ; U p qq H p M ; U p qq´ ´ H p M ; U p qq´ d where Ω ev d H - cl , Z p M q is the subgroup of those even forms on M which are twisted-closed and whose degree zerocomponent is given by an integer, i.e. ω “ n ` ω ` ω ` ¨ ¨ ¨ . Lemma 21 (The E -page for odd degrees in { AHSS ˆ h for twisted periodic Deligne cohomology) . The spectralsequence for the odd degrees looks as follows (5.2) 10 U p q ˆ Ω odd d H - cl p M q´ ´ H p M ; U p qq H p M ; U p qq´ ´ H p M ; U p qq where Ω odd d H - cl p M q is the group of twisted-closed odd forms on M . For twisted differential K -theory, in [GS17b] we identified the first nonzero differential in the spectralsequence as d p x q “ x Sq Z p x q ` r ˆ h s Y DB x , where x Sq Z is a torsion operation in differential cohomology inherited from Sq (see [GS16a]), and r ˆ h sY DB p´q is the Deligne-Beilinson cup product operation. The same argument used in [GS17b, Proposition 25] appliesto the case of differential refinements of the higher degree twists for periodic integral cohomology. As a resultwe have the following. Proposition 22 (First differential for { AHSS ˆ h for twisted Deligne cohomology) . Let h : M Ñ B k U p q ∇ bea twist of periodic Deligne cohomology. Then the differential in the associated AHSS on the E k ` -page is given by d k ` p x q “ ´r ˆ h s Y DB x . We will illustrate this in Examples 11 and 12 below. Note that there is also a differential on the E k -page; but we do not use this. .2. Examples via the spectral sequence.
We now proceed with our examples illustrating the AHSS thatwe developed in Section 5.1 to both twisted periodic integral cohomology (Section 2) and twisted periodicDeligne cohomology (Section 4).
Example 9 (Twisted periodic integral cohomology of even spheres) . For even spheres, the class of the twistvanishes for parity reasons. Therefore, the AHSS degenerates at the E -page and we immediately identify H ev p S k ; Z r u, u ´ sq “ Z ‘ Z and H odd p S k , Z r u, u ´ sq “ . Example 10 (Twisted periodic integral cohomology of odd spheres) . For an odd-dimensional sphere S k ` ,the only interesting twist occurs in degree k ` . Consequently, the only nonzero differential in the AHSSoccurs on the E k ` -page, giving the sequence Z ´r h sY / / H k ` p S k ` : Z q – Z / / . Thus, with h also denoting the integer corresponding to the topological twist h , we get H odd p S k ` ; Z r u, u ´ sq – Z { h and H ev p S k ` ; Z r u, u ´ sq – . The above examples illustrate the utility of the AHSS in computing the twisted integral cohomology. Wewill now extend these examples to the differential case. It turns out that for even spheres, we will find thatthe use of the Mayer-Vietrois sequence is straightforward enough and efficient in this case.
Example 11 (Twisted periodic Deligne cohomology of even spheres) . Let ˆ h : M Ñ B k U p q ∇ be atwist for periodic Deligne cohomology. For parity reasons, the class of the underlying topological twist h P H k ` p S k ; Z q vanishes. By property (iii) of Proposition 18, it follows that we have an isomorphism p H ev { odd p S k ; ˆ h q – p H ev { odd p S k ; Z r u, u ´ sq with the underlying untwisted theory. We computed the corresponding groups earlier in Example 4 (Section3.2), which immediately yields p H ev p S k ; ˆ h q “ Ω odd p S k q{ im p d q ‘ Z ‘ Z and p H odd p S k , ˆ h q “ Ω ev p S k q{ Ω evcl , Z p S k q . The case of odd spheres is more involved.
Example 12 (Twisted periodic Deligne cohomology of odd spheres) . For the odd spheres, the only inter-esting twist are the differential refinements of the topological twists r h s P H k ` p S k ` ; Z q . Choose such adifferential refinement ˆ h : M Ñ B k U p q ∇ . Then the spectral sequence has one nontrivial differential d k ` : U p q ˆ Ω odd d H - cl p S k ` q / / U p q – H k ` p S k ` ; U p qq , occurring on the p k ` q -page. Here Ω odd d H - cl denotes those odd forms which are closed under the twisteddifferential d H . The restriction to the factor U p q is given by the Deligne-Beilinson cup product with r ˆ h s .This can be computed as follows. As above, let h be the integer representing the topological class h P Z . For θ P U p q , we have r ˆ h s Y DB θ “ hθ . The kernel of d k ` restricted to this factor is the subgroup of h -rootsof unity which is isomorphic to Z { h . Since the map θ ÞÑ hθ is surjective the First Isomorphism Theoremimplies that the factor Ω odd d H - cl p S k ` q is killed by d k ` .It remains to solve the extension problem / / Z { h / / p H odd p S k ` ; ˆ h q / / Ω odd d H - cl p S k ` q / / . Now for any abelian group A and any divisible group B the Ext group Ext p A, B q vanishes. Since the group Ω odd d H - cl p S k ` q is divisible, we then have Ext ` Z { h, Ω odd d H - cl p S k ` q ˘ – Ω odd d H ´ cl p S k ` q{ h Ω odd d H - cl p S k ` q – . Thus, the extension must be the trivial one and we conclude that p H odd p S k ` ; ˆ h q – Z { h ‘ Ω odd d H - cl p S k ` q . Acknowledgement.
The authors thank the organizers and participants of the Geometric Analysis andTopology Seminar at the Courant Institute at NYU for asking about twisting Deligne cohomology, during atalk by H.S., which encouraged the authors to finish this two-stage project, starting with [GS17b]. eferences [ABG10] M. Ando, A. J. Blumberg, and D. J. Gepner, Twists of K-theory and TMF , Superstrings, geometry, topology, and C ˚ -algebras, 27–63, Proc. Sympos. Pure Math., 81, Amer. Math. Soc., Providence, RI, 2010, [ arXiv:1002.3004 ] [ math.AT ].[ABGHR14] M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk, Units of ring spectra, orientations and Thomspectra via rigid infinite loop space theory , J. Topol. (2014), no. 4, 1077–1117.[AH62] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces , 1961 Proc. Sympos. Pure Math. vol. III, pp.7–38, American Math. Soc., Providence, R.I., 1962.[AS06] M. Atiyah and G. Segal,
Twisted K-theory and cohomology , Inspired by S. S. Chern, 5-43, Nankai Tracts Math., 11,World Sci. Publ., Hackensack, NJ, 2006.[BB14] C. B¨ar and C. Becker,
Differential characters , Lecture Notes in Mathematics 2112, Springer, Cham, Switzerland, 2014.[Be85] A. Beilinson,
Higher regulators and values of L-functions , J. Soviet Math. (1985), 2036-2070.[BCMMS02] P. Bouwknegt, A. Carey, V. Mathai, M. Murray and D. Stevenson, Twisted K-theory and K-theory of bundlegerbes , Comm. Math. Phys. (2002), 17–45,[ arXiv:hep-th/0106194 ].[BT82] R. Bott and L. W. Tu,
Differential forms in algebraic topology , Springer-Verlag, New York-Berlin, 1982.[Br93] J.-L. Brylinski,
Loop spaces, characteristic classes and geometric quantization , Progress in Math. 107, Birkh¨auser,Boston, 1993.[BM94] J.-L. Brylinski and D. McLaughlin,
The geometry of degree-four characteristic classes and of line bundles on loopspaces I , Duke Math. J. (1994), no. 3, 603-638.[BM96] J.-L. Brylinski and D. McLaughlin, ˇCech cocycles for characteristic classes , Comm. Math. Phys. (1996), 225-236.[BFGM03] M. Bullejos, E. Faro, and M. A. Garcia-Mu˜noz, Homotopy colimits and cohomology with local coefficients , Cah.Topol. G´eom. Diff´er. Cat´eg. (2003), no. 1, 63-80.[Bu12] U. Bunke, Differential cohomology , [ arXiv:math.AT/1208.3961 ].[BKS10] U. Bunke, M. Kreck, and T. Schick,
A geometric description of differential cohomology , 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 sheaves of spectra , J. Homotopy Relat.Struct. (2016), no. 1, 1–66.[BS10] U. Bunke and T. Schick, Uniqueness of smooth extensions of generalized cohomology theories , J. Topol. (2010)110–156.[BSS07] U. Bunke, T. Schick and M. Spitzweck, Sheaf theory for stacks in manifolds and twisted cohomology for S -gerbes ,Algebr. Geom. Topol. (2007), 1007-1062.[CH89] J. Carlson and R. Hain, Extensions of variations of mixed Hodge structure. , Ast´erisque (1989), 9, 39–65.[CJMSW05] A. L. Carey, S. Johnson, M. K. Murray, D. Stevenson, and B.-L. Wang,
Bundle gerbes for Chern-Simons andWess-Zumino-Witten theories , Comm. Math. Phys. (2005), 577-613.[CS85] J. Cheeger and J. Simons,
Differential characters and geometric invariants , Lecture Notes in Math. , 50–80,Springer, Berlin, 1985.[De71] P. Deligne,
Th´eorie de Hodge: II , Pub. Math. IHES (1971), 5-57.[DL05] J. L. Dupont and R. Ljungmann, Integration of simplicial forms and Deligne cohomology , Math. Scand. (2005),11–39.[EV88] H. Esnault and E. Viehweg, Deligne-Beilinson cohomology , Beilinson’s Conjectures on Special Values of L-Functions,Academic Press, Boston, MA, 1988, 43–91.[FSS13] D. Fiorenza, H. Sati, and U. Schreiber,
Extended higher cup-product Chern-Simons theory , J. Geom. Phys. (2013),130–163, [ arXiv:1207.5449 ] [ hep-th ].[FSS14] D. Fiorenza, H. Sati, and U. Schreiber, Multiple M5-branes, string 2-connections, and 7d nonabelian Chern-Simonstheory , Adv. Theor. Math. Phys. (2014), no. 2, 229-321, [ arXiv:1201.5277 ] [hep-th].[FSS15a] D. Fiorenza, H. Sati, and U. Schreiber, A Higher stacky perspective on Chern-Simons theory , Mathematical Aspectsof Quantum Field Theories (Damien Calaque and Thomas Strobl eds.), Springer, Berlin (2015), [ arXiv:1301.2580 ] [ hep-th ].[FSS15b] D. Fiorenza, H. Sati, and U. Schreiber,
The E moduli 3-stack of the C -field in M-theory , Comm. Math. Phys. (2015), no. 1, 117-151. [ arXiv:1202.2455 ] [hep-th].[FSSt12] D. Fiorenza, U. Schreiber, and J. Stasheff, ˇCech cocycles for differential characteristic classes – An infinity-Lietheoretic construction , Adv. Theor. Math. Phys. (2012), 149–250, [ arXiv:1011.4735 ] [ math.AT ].[Fr00] D. S. Freed, Dirac charge quantization and generalized differential cohomology , Surv. Differ. Geom. 7, 129-194, Int.Press, Somerville, MA, 2000.[Fr01] D. S. Freed,
The Verlinde algebra is twisted equivariant K-theory , Turk. J. Math. (2001), 159–167.[Fr02] D. S. Freed, Classical Chern-Simons theory II , Special issue for S. S. Chern, Houston J. Math. (2002), no. 2, 293-310.[FH00] D. S. Freed and M. Hopkins, On Ramond-Ramond fields and K-theory , J. High Energy Phys. (2000), 44, 14 pp.[FL10] D. S. Freed and J. Lott, An index theorem in differential K-theory , Geom. Topol. (2010), no. 2, 903-966.[Ga97] P Gajer, Geometry of Deligne cohomology , Invent. Math. (1997) 155–207.[GPT13] L. Gallot, E. Pilon, and F. Thuillier,
Higher dimensional abelian Chern-Simons theories and their link invariants , J.Math. Phys. (2013), no. 2, 022305, 27 pp.[Gi84] H. Gillet, Deligne homology and Abel-Jacobi maps , BuIl. Amer. Math. Soc. (1984), 285-288.[Go01] K. Gomi, The formulation of the Chern-Simons action for general compact Lie groups using Deligne cohomology , J.Math. Sci. Univ. Tokyo (2001), no. 2, 223-242. Go06] K. Gomi,
Central extensions of gauge transformation groups of higher abelian gerbes , J. Geom. Phys. (2006), no. 9,1767-1781.[GT10] K. Gomi and U. Terashima, Chern-Weil construction for twisted K-theory , Comm. Math. Phys. (2010), no. 1,225-254.[GS16a] D. Grady and H. Sati,
Primary operations in differential cohomology , [ arXiv:1604.05988 ] [math.AT].[GS16b] D. Grady and H. Sati,
Spectral sequences in smooth generalized cohomology , Algebr. Geom. Topol. (2017), no. 4,2357-2412, [ arXiv:1605.03444 ] [math.AT].[GS17a] D. Grady and H. Sati, Twisted smooth Deligne cohomology , Ann. Glob. Anal. Geom. (2017), https://doi.org/10.1007/s10455-017-9583-z , [ arXiv:1706.02742 ] [math.DG].[GS17b] D. Grady and H. Sati,
Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral se-quence , [ arXiv:1711.06650 ] [math.AT].[Ha15] R. Hain,
Deligne-Beilinson cohomology of affine groups , [ arXiv:1507.03144 ] [math.AG].[HL06] R. Harvey and B. Lawson,
From sparks to grundles – differential characters , Comm. Anal. Geom. (2006), no. 1,25-58.[HLZ03] F. R. Harvey, H. B. Lawson, Jr. and J. Zweck, The deRham-Federer theory of differential characters and characterduality , Amer. J. Math. (2003), 791-847.[Hi01] N. Hitchin, Lectures on special Lagrangian submanifolds, in Winter School on Mirror Symmetry, Vector Bundles andLagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI,(2001), 151-182, [ arXiv:math.DG/9907034 ].[HQ15] M. J. Hopkins and G. Quick,
Hodge filtered complex bordism , J. Topology (2015), 147-183.[HS05] M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory , J. Differential Geom. (3)(2005), 329–452.[Ja88] U. Jannsen, Deligne homology, Hodge-D-conjecture, and motives , Beilinson’s Conjectures on Special Values of L-Functions, Academic Press, Boston, MA, 1988, 305-372.[Ja15] J. F. Jardine,
Local Homotopy Theory , Springer, New York, 2015.[Jo08] A. Joyal,
Notes on Logoi , preprint 2008, [ „ may/IMA/JOYAL/Joyal.pdf ][LSW16] J. A. Lind, H. Sati, and C. Westerland, Twisted iterated algebraic K-theory and topological T-duality for spherebundles , [ arXiv:1601.06285 ] [math.AT].[Lo02] J. Lott,
Higher degree analogs of the determinant line bundle , Comm. Math. Phys. (2002), 41-69.[Lu09] J. Lurie,
Higher topos theory , Princeton University Press, Princeton, NJ, 2009.[MW11] V. Mathai and S. Wu,
Analytic torsion for twisted de Rham complexes , J. Differential Geom. (2011), no. 2, 297–332,[ arXiv:0810.4204 ] [ math.DG ].[MQRT77] J. P. May, E ring spaces and E ring spectra , with contributions by F. Quinn, N. Ray, and J. Tornehave,Springer-Verlag, Berlin-New York, 1977.[MS06] J. P. May and J. Sigurdsson, Parametrized homotopy theory , Amer. Math. Soc., Providence, RI, 2006.[Pi05] R. Picken,
A cohomological description of abelian bundles and gerbes , Twenty years of Bialowieza: a mathematicalanthology, 217-228, World Sci. Monogr. Ser. Math., 8, World Sci. Publ., Hackensack, NJ, 2005, [ arXiv:math/0305147 ][math.DG].[Pr06] V. V. Prasolov,
Elements of Combinatorial and Differential Topology , Amer. Math. Soc., Providence, RI, 2006.[RW86] R. Rohm and E. Witten,
The antisymmetric tensor field in superstring theory , Ann. Physics (1986), no. 2, 454-489.[Ro82] J. Rosenberg,
Homological invariants of extensions of C ˚ -algebras , Operator algebras and applications, Part 1(Kingston, Ont., 1980), Proc. Sympos. Pure Math. , Amer. Math. Soc., Providence, RI, 1982, pp. 35–75.[Ro89] J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view , J. Austral. Math. Soc. Ser. A (1989),368–381.[Sa09] H. Sati, A higher twist in string theory , J. Geom. Phys. (2009), no. 3, 369-373, [ arXiv:hep-th/0701232 ].[Sa10] H. Sati, Geometric and topological structures related to M-branes , Superstrings, geometry, topology, and C ˚ -algebras,181–236, Proc. Sympos. Pure Math., 81, Amer. Math. Soc., Providence, RI, 2010, [ arXiv:1001.5020 ] [math.DG].[Sa14] H. Sati, M-Theory with framed corners and tertiary index invariants , SIGMA (2014), 024, 28 pages,[ arXiv:1203.4179 ] [hep-th].[SSS12] H. Sati, U. Schreiber, and J. Stasheff, Differential twisted String- and Fivebrane structures , 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].[Sh07] B. Shipley, H Z -algebra spectra are differential graded algebras , Amer. J. Math. (2) (2007), 351–379.[Sc13] U. Schreiber, Differential cohomology in a cohesive infinity-topos , [ arXiv:1310.7930 ] [math-ph].[SS08] J. Simons and D. Sullivan,
Axiomatic characterization of ordinary differential cohomology
J. Topol. (1) (2008), 45–56.[Te04] C. Teleman, K-theory and the moduli space of bundles on a surface and deformations of the Verlinde algebra , Topology,geometry and quantum field theory, 358-378, Cambridge Univ. Press, Cambridge, 2004.[Th15] F. Thuillier,
Deligne-Beilinson cohomology in U p q Chern-Simons theories , Mathematical aspects of quantum fieldtheories, 233-271, Math. Phys. Stud., Springer, Cham, 2015.[Wa13] K. Waldorf,
String connections and Chern-Simons theory , Trans. Amer. Math. Soc. (2013), 4393-4432,[ arXiv:0906.0117 ] [math.DG].] [math.DG].