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twisted cohomology

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cohomology

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Idea

By the discussion at cohomology, plain cohomology is the study of

and maybe fully generally in any (∞,1)-category 𝒞\mathcal{C} whatsoever.

So for A𝒞A \in \mathcal{C} any object the cohomology of any other object XX with coefficients in AA is the mapping space 𝒞(X,A)\mathcal{C}(X,A). Notice that this is equivalently the homotopy type of sections 𝒞 /X(X,X×A)\mathcal{C}_{/X}(X, X \times A) of the trivial AA-fiber ∞-bundle over XX. The idea of twisted cohomology then is to consider general AA-fiber ∞-bundles χ\chi over XX and take the χ\chi-twisted cohomology of XX to the type of sections of this.

cohomologytwisted cohomology
homotopy types of mapping spaceshomotopy types of spaces of sections

Given an \infty-topos H\mathbf{H}, then also its arrow \infty-category H I\mathbf{H}^I is an \infty-topos, over Grpd I\infty Grpd^I and it also sits over H\mathbf{H} by the codomain fibration, constituting an “extension” of H\mathbf{H} by itself:

H incl H I cod H. \array{ \mathbf{H} \\ \downarrow^{\mathrlap{incl}} \\ \mathbf{H}^I \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.

The intrinsic cohomology of H I\mathbf{H}^I under this fibration is nonabelian twisted cohomology as discussed in some detail in Principal ∞-bundles -- theory, presentations and applications.

Notice that “stable cohomology”, which is traditionally called generalized (Eilenberg-Steenrod) cohomology may be thought of as the lowest order Goodwillie approximation to nonabelian cohomology: where a cocycle in nonabelian cohomology is a map to any homotopy type, a cocycle in generalized (Eilenberg-Steenrod) cohomology is a map into a stable homotopy type.

In this sense the tangent (infinity,1)-topos THT \mathbf{H} is the lowest order linear approximation to the codomain fibration

Stab(H) incl TH cod H. \array{ \Stab(\mathbf{H}) \\ \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} \\ \downarrow^{\mathrlap{cod}} \\ \mathbf{H} } \,.

Higher-order approximations should involve a notion of higher-order forms of the tangent (∞,1)-topos, in parallel with the relationship between the jet bundles and tangent bundle of a manifold. It is clear that whatever we may say in detail about the kkth-jet (∞,1)-topos J kHJ^k \mathbf{H}, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.

It seems that a layered analysis of nonabelian cohomology this way in higher homotopy theory should eventually be rather important, even if it hasn’t received any attention at all yet. It seems plausible that a generalization of Chern-Weil theory which approximates classes of principal infinity-bundles not just by universal characteristic classes in ordinary cohomology and hence in stable cohomology, but that one wants to consider the whole Goodwillie Taylor tower of approximations to it.

Definition

We discuss concrete realizations of the above general idea in some cases of interest:

In an \infty-topos – twisted nonabelian (sheaf) cohomology

Let 𝒞=H\mathcal{C} = \mathbf{H} be an (∞,1)-topos. Let AHA \in \mathbf{H} be any object, to be called the coefficient object.

Write Aut(A)Grp(H)\mathbf{Aut}(A) \in Grp(\mathbf{H}) for the automorphism ∞-group of AA and BAut(A)H\mathbf{B}\mathbf{Aut}(A) \in \mathbf{H} for its delooping. There is a canonical ∞-action of Aut(A)\mathbf{Aut}(A) on AA exhibited by the corresponding universal associated ∞-bundle

A A//Aut(A) ρ A BAut(A). \array{ A &\to& A//\mathbf{Aut}(A) \\ && \downarrow^{\mathrlap{\rho_A}} \\ && \mathbf{B}\mathbf{Aut}(A) } \,.

Let XHX \in \mathbf{H} be any object.

Definition

A twist for AA-cohomology on XX is a morphism χ:XBAut(A)\chi \colon X \to \mathbf{B}\mathbf{Aut}(A) in H\mathbf{H}. The corresponding associated AA-fiber ∞-bundle over XX which is the homotopy pullback

χ *ρ A A//Aut(A) ρ A X χ BAut(A) \array{ \chi^\ast \rho_A &\to& A//\mathbf{Aut}(A) \\ \downarrow && \downarrow^{\mathrlap{\rho_A}} \\ X &\stackrel{\chi}{\to}& \mathbf{B}\mathbf{Aut}(A) }

we call the local coefficient ∞-bundle for twisted AA-cohomology classified by χ\chi.

The cocycle ∞-groupoid of χ\chi-twisted AA-cohomology is

Γ X(χ *ρ A)Grpd. \Gamma_X(\chi^\ast \rho_A) \in \infty Grpd \,.

The χ\chi-twisted cohomology set of XX is

π 0Γ X(χ *ρ A)Set \pi_0 \Gamma_X(\chi^\ast \rho_A) \in Set

Special cases of this definition are implicit in traditional literature. The above statement appears in this form in (Nikolaus-Schreiber-Stevenson 12).

Remark

The χ\chi-twisted cohomology is equivalently the ordinary cohomology of χ\chi with coefficients in ρ A\rho_A in the slice (∞,1)-topos of H\mathbf{H} over BAut(A)\mathbf{B}\mathbf{Aut}(A):

Γ X(χ *ρ A)H /BAut(A)(χ,ρ A). \Gamma_X(\chi^\ast \rho_A) \simeq \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(A)}(\chi, \rho_A) \,.

In a stabilized \infty-topos – twisted ES-type (sheaf) cohomology

Let now 𝒞=Stab(H)\mathcal{C} = Stab(\mathbf{H}) be an stable (∞,1)-category of spectrum objects in an ambient (∞,1)-topos H\mathbf{H}. Let ECRing (H)E \in CRing_\infty(\mathbf{H}) be a corresponding E-∞ ring object. Write

GL 1(E)Aut(E)Grp(H) GL_1(E) \hookrightarrow \mathbf{Aut}(E) \in Grp(\mathbf{H})

for the ∞-group of units of EE.

Now a twist χ:XBGL 1(E)\chi \;\colon\; X \to \mathbf{B}GL_1(E) classifies an (∞,1)-module bundle of EE-lines. The χ\chi-twisted EE-cohomology is again the (stable) homotopy type of sections of this.

For the case of twisted K-theory (see the references there) this description goes back to Jonathan Rosenberg. The above general abstract description is developed in (Ando-Blumberg-Gepner 10).

For more details see

Remark

There are canonical maps

BGL 1(E)ELinePic(EMod)EMod, \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow Pic(E Mod) \hookrightarrow E Mod \,,

where Pic(EMod)Pic(E Mod) denotes the Picard ∞-groupoid. This suggest to speak not just of twists of the form χ:XBGL 1(E)ELineEMod\chi \colon X \to \mathbf{B}GL_1(E) \simeq E Line \hookrightarrow E Mod but more generally of twists of the form χ:Pic(EMod)EMod\chi \colon Pic(E Mod) \hookrightarrow E Mod. While these in general no longer define EE-fiber ∞-bundles (so that sections of them are strictly speaking in general no longer locally EE-cohomology cocycles), this more general notion has the advantage that it makes sense also in symmetric monoidal (∞,1)-categories different from those of the form Stab(H)Stab(\mathbf{H}).

This we turn to below.

In a general symmetric monoidal \infty-category

Remark

If in the above situation we write [X,EMod][X, E Mod] for the symmetric monoidal (∞,1)-category of EE-(∞,1)-module bundles on XX, then given an object χ[X,EMod]\chi \in [X,E Mod] its homotopy type of sections, hence the χ\chi-twisted cohomology of XX is equivalently

Hom [X,EMod](𝕀 X,χ)Grpd, Hom_{[X, E Mod]}(\mathbb{I}_X, \chi) \in \infty Grpd \,,

where 𝕀 X\mathbb{I}_X is the tensor unit object, the trivial EE-(∞,1)-module bundle over XX.

In view of this and remark 2 one considers the following.

Let (𝒞,)(\mathcal{C}, \otimes) be a symmetric monoidal (∞,1)-category.

Definition

An object χPic(𝒞)\chi \in Pic(\mathcal{C}) of the Picard ∞-groupoid of 𝒞\mathcal{C} we call a twist for cohomology in 𝒞\mathcal{C}. For X,A𝒞X, A \in \mathcal{C} any two objects, we say that the χtwisted\chi-twisted cohomology of XX with coefficients in AA is

𝒞(X,χA)Grpd. \mathcal{C}(X, \chi \otimes A) \in \infty Grpd \,.

Properties

Twisted cohomology with trivial twisting cocycle

old material, to be harmonized

Let *B{*} \to B be a pointed object. Then

  • we say that the cocycle

    (X*B)H(X,B)(X \to * \to B) \in \mathbf{H}(X,B)

    is the trivial BB-cocycle on XX.

  • the morphism f:B^Bf:\hat{B}\to B induces a fibration sequence

    AB^fBA \to \hat B \stackrel{f}{\to} B

    in H\mathbf{H}.

Proposition

The ([*],f)([*],f)-twisted cohomology with trivial twisting cocycle is equivalent to the ordinary cohomology with coefficients in the homotopy fiber AA of ff:

H [*](X,f)H(X,A). \mathbf{H}_{[*]}(X,f) \simeq \mathbf{H}(X,A) \,.
Proof

By definition, the homotopy fiber of AA is the homotopy pullback

A * B^ f B \array{ A &\to& * \\ \downarrow && \downarrow \\ \hat B &\stackrel{f}{\to}& B }

in H\mathbf{H}. Since the \infty-groupoid valued hom in an (∞,1)-category is exact with respect ot homotopy limits (by definition of homotopy limits), it follows that for every object XX, there is fibration sequence of cocycle ∞-groupoids

H(X,A) * const * H(X,B^) H(X,B). \array{ \mathbf{H}(X,A) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{const_{*}}} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,.

By definition of twisted cohomology, this identifies

H(X,A)H [*](X,f). \mathbf{H}(X,A) \simeq \mathbf{H}_{[*]}(X,f) \,.

For this reason, when BB is pointed, it is customary to call the set of equivalence classes π 0H [c](X;f)\pi_0\mathbf{H}_{[c]}(X;f) the cc-twisted AA-cohomology of XX, and to denote it by the symbol

H [c](X,A) H_{[c]}(X,A)
Remark

The cohomology fibration sequence H(X,A)H(X,B^)H(X,B)\mathbf{H}(X,A) \to \mathbf{H}(X,\hat B) {\to} \mathbf{H}(X,B) can be seen as an obstruction problem in cohomology:

  • the obstruction to lifting a B^\hat B-cocycle to an AA-cocycle is its image in BB-cohomology (all with respect to the given fibration sequence)

But it also says:

  • AA-cocycles are, up to equivalence, precisely those B^\hat B-cocycles whose class in BB-cohomology is the class of the trivial BB-cocycle.

Examples

Sections as twisted functions

For VV a vector space and XX a manifold, both regarded a 0-truncated objects in the (,1)(\infty,1)-topos on the site CartSp (that of Lie infinity-groupoids), a cocycle XVX \to V is simply smooth VV-valued function on XX.

Now let GG be a Lie group with smooth delooping groupoid BG\mathbf{B}G and let ρ:BGVect\rho : \mathbf{B}G \to Vect be a representation of GG on VV, i.e. ρ()=V\rho(\bullet) = V. Then the corresponding action groupoid V//GV//G sits in the fibration sequence

VV//GpBG. V \to V//G \stackrel{p}{\to} \mathbf{B}G \,.

Hence we can ask for the pp-twisted cohomology of XX with values in VV. Now, a cocycle g:XBGg : X \to \mathbf{B}G is one classifying a GG-principal bundle on XX. By looking at this in Cech cohomology it is immediate to convince onself that cocycles XV//GX \to V//G such that the composite XV//GpBGX \to V//G \stackrel{p}{\to} \mathbf{B}G is equivalent to the given gg are precisely the sections of the ρ\rho-associated vector bundle:

on a patch U iU_i of a good cover over wich PP has been trivialized, the cocycle XV//GX \to V//G is simply a VV-valued function σ i:U iV\sigma_i : U_i \to V. Then on double overlaps it is a smooth natural transformation σ i U ijσ j U ij\sigma_i|_{U_{i j}} \to \sigma_j|_{U_i j} whose components in GG are required to be those of the given cocycle gg. That means exactly that the functions (σ i)(\sigma_i) are glued on double overlaps precisely as the local trivializations of a global section σ:XP× GV\sigma : X \to P \times_G V would.

Hence we find the pp-twisted cohomology is

H [g](X,V)={sectionsofP× GV}. H_{[g]}(X,V) = \{sections\; of\; P \times_G V\} \,.

In this sense a section is a twisted function.

Notice that V//GpBGV//G \stackrel{p}{\to} \mathbf{B}G is not itself a homotopy fiber, but is a lax fiber, in that we have a lax pullback (really a comma object )

V//G * BG Vect, \array{ V//G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& Vect } \,,

where in the bottom right corner we have Vect (regarded as a stack on CartSpCartSp in the evident way) and where the right vertical morphism sends the point to the ground field vector space kk (or rather sends UCartSpU \in CartSp to the trivial vector bundle U×kU \times k ).

We may paste to this the homotopy pullback along the cocycle g:XBGg : X \to \mathbf{B}G to obtain

P× GV V//G * X g BG Vect. \array{ P\times_G V &\to& V//G &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G &\to& Vect } \,.

This makes is manifest that a section σ:XP× GV\sigma : X \to P \times_G V is also the same as a natural transformation from const k:XVectconst_k : X \to Vect to XgBGVectX \stackrel{g}{\to} \mathbf{B}G \to Vect.

Notice moreover that in the special case that G=U(1)G = U(1) and for ground field k=k = \mathbb{C} we may think of BU(1)\mathbf{B}U(1) as the category LineMod=Vect\mathbb{C} Line \hookrightarrow \mathbb{C} Mod = Vect and think of the twisting cocycle gg as

XgLineMod. X \stackrel{g}{\to} \mathbb{C}Line \hookrightarrow \mathbb{C}Mod \,.

… and \infty-sections as twisted \infty-functions

Regarded this way, the above discussion has a generalization to the case where the monoid \mathbb{C} is replaced with any ring spectrum RR and we consider

XτRLineRMod. X \stackrel{\tau}{\to} R Line \hookrightarrow R Mod \,.

Twisted cohomology in terms of such morphisms τ\tau is effectively considered in

and in unpublished work of Ulrich Bunke et al. For more on this see the discussion at (∞,1)-vector bundle.

More generally one can hence twist with maps

XPic(R)RMod X \to Pic(R) \hookrightarrow R Mod

into the Picard ∞-group of RModR Mod.

See also at ∞-group of units – augmented definition.

twisted K-theory

In the context of generalized (Eilenberg–Steenrod) cohomology a coefficient object for cohomology is a spectrum AA: the AA-cohomology of a topological space XX with coefficients in AA is the set of homotopy classes of maps XAX \to A. For instance, as a model of the degree-00 space in the K-theory spectrum one can take A=Fred(H)A = Fred(H), the space of Fredholm operators on a separable Hilbert space HH. There is a canonical action on this space of the projective unitary group G=PU(H)G = P U(H) of HH. Since PU(H)P U(H) has the homotopy type of an Eilenberg–Mac Lane space K(,2)K(\mathbb{Z},2), a PU(H)P U(H)-principal bundle PXP \to X defines a class cH 3(X,)c \in H^3(X,\mathbb{Z}) in ordinary integral cohomology (this may also be thought of as the class of a twisting bundle gerbe). The twisted K-theory (in degree 00) of XX with that class as its twist is the set of homotopy classes of sections XP× PU(H)Fred(H)X \to P \times_{P U(H)} Fred(H) of the associated bundle.

GG-Actions on spectra

The above example generalizes straightforwardly to the case that

  • AA is a connective spectrum, i.e. topological space that is an infinite loop space. (We need to assume a connective spectrum given by an infinite loop space so that AA can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces BG\mathbf{B}G of nonabelian groups);

  • with a (topological) group GG acting on AA by automorphisms and

  • a GG-principal bundle PX.P \to X.

In this case there is an established definition of generalized (Eilenberg–Steenrod) cohomology with coefficients AA twisted by a GG-principal bundle as follows.

From the GG-principal bundle PXP \to X we obtain the associated AA-bundle P× GAXP \times_G A \to X. Then a twisted AA-cocycle on XX is defined to be a section XP× GAX \to P \times_G A of this associated bundle. The collection of homotopy classes of these sections is the twisted AA-cohomology of XX with the twist specified by the class of PP.

This is the definition of twisted cohomology as it appears for instance essentially as definition 22.1.1 of the May–Sigursson reference below (when comparing with their definition take their GG to be the trivial group and identify their Γ\Gamma and Π\Pi with our GG).

It is clearly a particular case of the general definition of twisted cohomology given above:

  • the (,1)(\infty,1)-topos H\mathbf{H} is the (,1)(\infty,1)-category of Top of topological spaces

  • the object BB is the delooping BG\mathbf{B}G of the group GG.

  • the object B^\hat{B} is the homotopy quotient A//GEG× GAA//G\simeq \mathbf{E}G\times_G A.

  • the twisting cocycle cc is the element in Top(X,BG)\mathbf{Top}(X,\mathbf{B}G) defining the principal GG-bundle PXP\to X.

Indeed, BB is pointed, we have a fibration sequence

AA//GBG A \to A//G \to \mathbf{B}G

and the homotopy pullback

P A A//G f X c BG \array{ P_A &\to& {A//G} \\ \downarrow && \downarrow{f} \\ X &\stackrel{c}{\to}& \mathbf{B}G }\,

is the AA-bundle P× GAXP\times_G A\to X.

The obstruction problem described by this example reads as folllows:

  • the obstruction to lifting a (“nonabelian” or “twisted”) A//GA//G-cocycle g:XA//Gg : X \to A//G to an AA-cocycle g^:XA\hat g : X \to A is its image δg\delta g in first GG-cohomology δgH 1(X,G):=π 0Top(X,BG)\delta g \in H^1(X,G) :=\pi_0 \mathbf{Top}(X, \mathbf{B} G).

Read the other way round it says:

  • AA-cocycles are precisely those GG-twisted AA-cocycles whose twist vanishes.
Remark

Since the associated bundle P× GAP \times_G A is in general no longer itself a spectrum, twisted cohomology is not an example of generalized Eilenberg–Steenrod cohomology.

To stay within the spectrum point of view, May–Sigurdsson suggested that twisted cohomology should instead be formalized in terms of parameterized homotopy theory, where one thinks of P× GAP \times_G A as a parameterized family of spectra.

Group cohomology with coefficients in a module

Some somewhat trivial examples of this appear in various context. For instance group cohomology on a group with coefficients in a nontrivial module can be regarded as an example of twisted cohomology. See there for more details.

Compare this to the example below of cohomology “with local coefficients”. It is the same principle in both cases.

Twisted bundles

To get a feeling for how the definition does, it is instructive to see how for the fibration sequence coming from an ordinary central extension KG^GK \to \hat G \to G of ordinary groups as

BG^BGωB 2K \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\omega}{\to} \mathbf{B}^2 K

classified by a group 2-cocycle ω\omega, cc-twisted G^\hat G-cohomology produces precisely the familiar notion of twisted bundles, with the twist being the lifting gerbe that obstructs the lift of a GG-bundle to a G^\hat G-bundle.

This is also the first example in the list in the last section of

and contains examples that are of interest in the wider context of string theory.

See also Twisted Differential String- and Fivebrane-Structures.

Cohomology with local coefficients

What is called cohomology with local coefficients is twisted cohomology with the twist given by the class represented by the universal cover space of the base space, which is to say: by the action of the fundamental group of the base space.

In the classical case of ordinary cohomology, C. A. Robinson in 1972 constructed a twisted K(π,n)K(\pi,n), denoted K˜(π,n)\tilde K(\pi,n), so that, for nice spaces, the cohomology with local coefficients H˜ n(X,π)\tilde H^n(X,\pi) with respect to a homomorphism ε:π 1(X)Aut(π)\varepsilon:\pi_1(X)\to Aut(\pi) is given by homotopy classes of maps XK˜(π,n)X\to \tilde K(\pi,n) compatible with ε.\varepsilon.

More generally, for any space XX, let AA be a coefficient object that is equipped with an action of the first fundamental group π 1(X)\pi_1(X) of XX. (Such an action is also called an AA-valued local system on XX).

Then there is the fibration sequence

AA//π 1(X)Bπ 1(X) A \to A//\pi_1(X) \to \mathbf{B} \pi_1(X)

of this action.

Notice that there is a canonical map c:XBπ 1(X)c : X \to \mathbf{B} \pi_1(X), the one that classifies the universal cover of XX.

Then AA-cohomology with local coefficients on XX is nothing but the cc-twisted AA-cohomology of XX in the above sense.

Effective computation of cohomology with local coefficients

By effective, we mean involving as much as possible only calculations within finite dimensional linear algebra. For definiteness, we work in the smooth context and require the locally constant sheaf 𝒜\mathcal{A} to have stalks of finite dimensional vector spaces over a field kk (\mathbb{R} or \mathbb{C}). Let XX be a connected nn-dimensional manifold. The sheaf 𝒜\mathcal{A} can then be seen as the sheaf of germs of locally constant sections of a vector bundle AXA\to X endowed with a flat connection \nabla. The fibers of AA are isomorphic with the stalks of 𝒜\mathcal{A}, all of which are isomorphic to some finite dimensional vector space A¯\bar{A}. Let X˜X\tilde{X} \to X denote the universal cover of XX and π=π 1(X)\pi = \pi_1(X) its fundamental group. It is well known that π\pi acts by deck transformation diffeomorphisms on X˜\tilde{X} and also induces a holonomy representation ρ:πAut(A¯)\rho\colon \pi \to \mathbf{Aut}(\bar{A}) on A¯\bar{A}.

Consider the vector bundles Λ pX XA\Lambda^p X \otimes_X A, where Λ pX\Lambda^p X is the bundle of differential pp-forms, with Ω A p()\Omega^p_A(-) denoting the sheaf of its sections (differential forms twisted by AA). Let d :Ω A pΩ A p+1d_\nabla \colon \Omega^p_A \to \Omega^{p+1}_A denote the correspondingly twisted de Rham differential, defined by the property that

(1)d (ωa)=dωa+(1) ωωa, d_\nabla(\omega \otimes a) = d\omega \otimes a + (-1)^{|\omega|} \omega \wedge \nabla a ,

where dd is the ordinary de Rham differential, a\nabla a is seen as a section of Λ iX XA\Lambda^i X \otimes_X A, and with the wedge operation acting in the obvious way. The complex of sheafs (Ω A (),d )(\Omega^\bullet_A(-),d_\nabla) is then a soft sheaf? resolution of the sheaf 𝒜\mathcal{A} of locally constant sections of AA. Its cohomology H p(X;A,)=H p(Ω A (X),d )H^p(X;A,\nabla) = H^p(\Omega^\bullet_A(X),d_\nabla) is then isomorphic to the sheaf cohomology of XX with coefficients in the locally constant sheaf 𝒜\mathcal{A}, H p(X,𝒜)H p(X;A,)H^p(X,\mathcal{A}) \simeq H^p(X;A,\nabla).

Now, the bundle AXA\to X and the connection \nabla both pull back to the universal covering space X˜\tilde{X}, that is to A˜X\tilde{A} \to X and ˜\tilde{\nabla}. Since now X˜\tilde{X} is simply connected, we can globally trivialize this bundle as A˜X˜×A¯\tilde{A} \simeq \tilde{X} \times \bar{A} and ˜\tilde{\nabla} to the trivial connection thereon. Similarly, the structure of the sheaf of twisted differential forms can be simplified to Ω A˜ p()Ω p()A¯\Omega^p_{\tilde{A}}(-) \simeq \Omega^p(-) \otimes \bar{A}, with the action of the twisted de Rham differential given by d ˜(ωa¯)=dωa¯d_{\tilde{\nabla}} (\omega \otimes \bar{a}) = d\omega \otimes \bar{a}. This observation allows us to conclude that, on the universal covering space, we have the isomorphism H p(X˜;A˜,˜)H p(X˜)A¯H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}, where H p(X˜)H^p(\tilde{X}) denotes the ordinary de Rham cohomology.

The pull back along a deck transformation diffeomorphism, induces a linear action of π\pi on forms Ω p(X˜)\Omega^p(\tilde{X}). Combined with the holonomy representation of π\pi on A¯\bar{A}, this defines a representation of π\pi on Ω A˜ p(X)\Omega^p_{\tilde{A}}(X). Let Ω A˜ p(X˜) πΩ A˜ p\Omega^p_{\tilde{A}}(\tilde{X})^\pi \subseteq \Omega^p_{\tilde{A}} denote the subspace of twisted forms that is invariant under the action of π\pi. It is not hard to notice the isomorphism of complexes (Ω A˜ (X˜) π,d ˜)(Ω A (X),d )(\Omega^\bullet_\tilde{A}(\tilde{X})^\pi, d_{\tilde{\nabla}}) \simeq (\Omega^\bullet_A(X), d_\nabla) and hence of their cohomologies, H p(X;A,)H p(Ω A˜ (X˜) π,d ˜)H^p(X; A,\nabla) \simeq H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi, d_{\tilde{\nabla}}). Furthermore, the action of π\pi on Ω A˜ p(X˜)\Omega^p_{\tilde{A}}(\tilde{X}) commutes with the differential d ˜d_{\tilde{\nabla}} and hence induces an action on H p(X˜;A˜,˜)H p(X˜)A¯H^p(\tilde{X}; \tilde{A},\tilde{\nabla}) \simeq H^p(\tilde{X}) \otimes \bar{A}, where π\pi also acts in the obvious and compatible way on each tensor factor. Let (H p(X˜)A¯) π(H^p(\tilde{X})\otimes \bar{A})^\pi denote the corresponding π\pi-invariant subspace.

Proposition

Suppose that there exists a decomposition Ω A˜ p(X˜)Ω A˜ p(X˜) πΩ A˜ p(X˜) π^\Omega^p_{\tilde{A}}(\tilde{X}) \simeq \Omega^p_{\tilde{A}}(\tilde{X})^\pi \oplus \Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}} as representations of π\pi, with Ω A˜ p(X˜) π^\Omega^p_{\tilde{A}}(\tilde{X})^{\hat{\pi}} having no π\pi-invariant subspace. Then we have the following isomorphism for each pp: H p(X,𝒜)(H p(X˜)A¯) πH^p(X,\mathcal{A}) \simeq (H^p(\tilde{X})\otimes \bar{A})^\pi.

Whether the decomposition hypothesis actually holds may depend on the properties of the group π\pi. For instance, it does hold if π\pi is compact (finite, in particular). Other cases, have to be examined individually.

Proof

Start with the short exact sequence of complexes

(2)0Ω A˜ (X˜) πΩ A˜ (X˜)Ω A˜ (X˜) π^0. 0 \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^\pi \to \Omega^\bullet_{\tilde{A}}(\tilde{X}) \to \Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}} \to 0 .

The corresponding long exact sequence in cohomology is equivalent to the short exact sequences

(3)0H p(X,𝒜)H p(X˜)A¯H p(Ω A˜ (X˜) π^,d ˜)0 0 \to H^p(X,\mathcal{A}) \to H^p(\tilde{X}) \otimes \bar{A} \to H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}}) \to 0

for each value of pp. The reason that all the connecting maps in the long exact sequence are zero is representation theoretic, since all the relevant maps are π\pi-equivariant. Since, H p+1(X,𝒜)H^{p+1}(X,\mathcal{A}) carries a trivial representation by π\pi, while the H p(Ω A˜ (X˜) π^,d ˜)H^p(\Omega^\bullet_{\tilde{A}}(\tilde{X})^{\hat{\pi}}, d_{\tilde{\nabla}}) representation has no π\pi-invariant subspace, by Schur's lemma, the only equivariant map from the latter to the former is zero. From the same observation, we easily see that the inclusion of H p(X,𝒜)H^p(X,\mathcal{A}) must coincide with the π\pi-invariant subspace (H p(X˜)A¯) π(H^p(\tilde{X}) \otimes \bar{A})^\pi.

The presentation of cohomology of XX with local coefficients 𝒜\mathcal{A} as π\pi-invariant de Rham cohomology of the universal covering space X˜\tilde{X} twisted by the holonomy representation on the stalk A¯\bar{A} is originally due to (Eilenberg 47). It is also discussed in Chapter VI of (Whitehead 78). The idea to look at the π\pi-invariant subspace of the twisted de Rham cohomology of the universal covering space scan be found in an answer by Peter Michor on MathOverlflow.

The above result can be seen as an effective way to compute the sheaf cohomology groups H p(X,𝒜)H^p(X,\mathcal{A}) since all it requires is the knowledge of the following finite dimensional representations of the fundamental group π=π 1(X)\pi = \pi_1(X): the deck transformations on the de Rham cohomology H k(X˜)H^k(\tilde{X}) of the covering space, and the holonomy representation on a typical stalk A¯\bar{A} of the locally constant sheaf 𝒜\mathcal{A}. Obtaining the invariant subspace of their tensor product can then be done using usual representation theory methods, which involve only finite dimensional linear algebra. Unfortunately, it appears that the requirement that π\pi is finite is rather important for the argument. It is not entirely clearly how to proceed if, for instance π=\pi = \mathbb{Z} or is non-abelian and infinite.

References

General

A discussion of π 1(X)\pi_1(X)-twisted ordinary cohomology is in

  • M. Bullejos, E. Faro, M. A. García-Muñoz, Homotopy colimits and cohomology with local coefficients, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 (numdam)

For the special case of generalized (Eilenberg–Steenrod) cohomology twisted by a GG-principal bundle see section 22.1 of

This in turn is based on the definition of twisted K-theory given in

Details on Larmore’s work on twisted cohomology are at

The abstract discussion of twisted nonabelian cohomology in \infty-toposes is in

The abstract discussion of twisted ES-type cohomology in the stable (infinity,1)-category of spectra is in

The presentation of cohomology with local coefficients in terms π 1\pi_1-equivariant de Rham cohomology on the universal covering space is discussed in

  • S. Eilenberg, Homology of spaces with operators I, Trans. Amer. Math. Soc. 6 (1947), 378-417. (doi)
  • G. W. Whitehead, Elements of homotopy theory, Springer-Veriag, 1978.

Chronology of literature on twisted cohomology

The oldest meaning of twisted cohomology is that of cohomology with local coefficients (see above).

For more on the history of that notion see

In the following we shall abbreviate

  • tc = twisted cohomology

Searching MathSciNet for twisted cohomology led to the following chronology: It missed older references in which the phrase was not used but the concept was in the sense of local coefficient systems – ancient and honorable.

Most notably missing are

  • Kurt Reidemeister (1938) Topologie der Polyeder und kombinatorische Topologie der Komplexe_, Mathematik und ihre Anwendungen in Physik und Technik,_(But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)

  • Norman Steenrod (1942,1943)

  • Olum (thesis 1947, published 1950)

Next come several that involve twisted differentials more generally.

Few are in terms of homotopy of spaces

tc ops should be treated as a single phrase – it may be that the ops are twisted, not the cohomology

  • 1966 McClendon thesis – summarized in

  • 1967 Emery Thomas tc ops

  • 1967 Larmore tc ops

  • 1969 McClendon tc ops

  • 1969 Larmore tc

  • 1970 Peterson tc ops

  • 1971 McClendon tc ops

  • 1972 Deligne Weil conjecture for K3 tc – meaning?

  • 1972 Larmore tc

  • 1973 Larmore and Thomas tc

  • 1973 Larmore tc

  • gap

  • 1980 Coelho & Pesennec tc

  • 1980 Tsukiyama sequel to McClendon

  • 1983 Coelho & Pesennec tc

  • 1985 Morava but getsted at 1975 ??

  • 1986 Fried tc

  • 1988 Baum & Connes ??

  • 1989 Lott torsion

  • 1990 Dwork ??

  • 1993 Gomez–Tato tc minimal models

  • 1993 Duflo & Vergne diff tc

  • 1993 Vaisman tc and connections

  • 1993 Mimachi tc and holomorphic

  • 1994 Kita tc and intersection

  • 1995 Cho, Mimachi and Yoshida tc and configs

  • 1995 Cho, Mimachi tc and intersection

  • 1996 Iwaski and Kita tc de rham

  • 1996 Asada nc geom and strings

  • 1997 H Kimura tc de Rham and hypergeom

  • 1998 Michael Farber, Gabriel Katz?, Jerome Levine?, Morse theory of harmonic forms, Topology, (Volume 37, Issue 3, May 1998, Pages 469–483)

  • 1998 Knudson tc SL_n

  • 1998 Morita tc de Rham

  • 1999 Kachi, Mtsumoto, Mihara tc and intersection

  • 1999 Hanamura & Yoshida Hodge tc

  • 1999 Felshtyn & Sanchez–Morgado Reidemeister torsion

  • 1999 Haraoka hypergeom

  • 2000 Tsou & Zois tc de rham

  • 2000 Manea tc Czech

  • 2001 Royo Prieto tc Euler

  • 2001 Takeyama q-twisted

  • 2001 Gaberdiel &Schaefr–Nameki tc of Klein bottle

  • 2001 Iwaskai tc deRham

  • 2001 Proc Rims tc and DEs and several papers in this book

  • 2001 Knudson tc SL_n

  • 2001 Royo Prieto tc as d+kd+k\wedge

  • 2001 Barlewtta & Dragomir tc and integrability

  • 2002 Lueck L 2L^2

  • 2002 Verbitsky HyperKahler, torsion, etc

  • 2003 Etingof & Grana tc of Carter, Elhamdadi and Saito

  • 2003 Cruikshank tc of Eilenberg

  • 2003 various in Proc NATO workshop

  • 2003 Dimca tc of hyperplanes

  • 2004 Kirk & Lesch tc and index

  • 2004 Bouwknegt, Evslin, Mathai tc and tK

  • 2004 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

  • 2005 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

  • 2005 Bunke & Schick tc in re: T-duality

  • 2006 Dubois tc and Reidemeister (elsewhere he considers twisted Reidemesiter)

  • 2006 Bunke & Schick tc in re: T-duality

  • 2006 Sati

  • 2006 Atiyah & Segal tc and tK

  • 2007 Mickelsson & Pellonpaa tc and tK

  • 2007 Sugiyama in re: Galois and Reidemeister

  • 2007 Bunke, Schick, Spitzweck tc in re: gerbes

  • 2008 Kawahara hypersurfaces

Revised on April 1, 2014 04:16:33 by Urs Schreiber (185.37.147.12)