Domenico Fiorenza
twisted cohomology


Special and general types

Special notions


Extra structure



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For f:B^Bf:\hat B \to B a morphism in an (∞,1)-topos? H\mathbf{H}, twisted cohomology at stage XX is the fiber of ff over a given element? of BB.


We now say this again, more formally and more in detail. For

twisted cohomology

H [c](X,f):=π 0H c(X,f) H^{[c]}(X,f):=\pi_0\mathbf{H}^c(X,f)

is the set of equivalence classes of the fiber H c(X,f)\mathbf{H}^c(X,f) of the morphism f:B^Bf:\hat B \to B over the generalized element? element cc of BB. More explicitly, H c(X,f)\mathbf{H}^c(X,f) is the ∞-groupoid? defined as the homotopy pullback?

H c(X,f) * *c H(X,B^) H(X,B). \array{ \mathbf{H}^{c}(X,f) &\to& {*} \\ \downarrow && \;\downarrow^{{*}\mapsto c} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,.


In terms of sections

The cocycle cH(X,B)c\in\mathbf{H}(X,B) induces the homotopy pullback

P f B^ f X c B. \array{ P_f &\to& {\hat{B}} \\ \downarrow && \downarrow{f} \\ X &\stackrel{c}{\to}& B }\,.

By the universal property of homotopy pullback, homotopy classes of sections XP fX \to P_f, are in bijection with homotopy classes of homotopy commutative cones of the form

X id X c B f B^. \array{ && X \\ & {}^{id}\swarrow && \searrow \\ X &\stackrel{c}{\to}& B &\stackrel{f}\leftarrow& \hat{B} } \,.

These in turn are manifestly the homotopy classes of maps XB^X \to \hat{B} such that XB^BX \to \hat{B} \to B is homotopic to cc. So

π 0Γ(P f)H [c](X,f) \pi_0 \Gamma(P_f) \simeq H^{[c]}(X,f)

Twisted cohomology with a pointed base

If BB is pointed, then 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}, i.e. a homotopy pullback? square

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

with *{*} denoting the? point? (the? terminal object?).

Since the \infty-groupoid valued hom in an (∞,1)-category? is exact? with respect to 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^{const_{*}} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,,

so that H const *(X,f)H(X,A)\mathbf{H}^{const_{*}}(X,f)\simeq \mathbf{H}(X,A). In other words, twisted cohomology with the trivial twisting cocycle const *const_{*} is nothing but cohomology with coefficients in the homotopy fiber of ff over the distinguished point of BB. 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)

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 trivial class (given by the trivial cocycle const *:*const_{*} : {*} \to ).


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

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:

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:

Read the other way round it says:


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 \hat Gbundle-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.


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 above definition of cc-twisted cohomology as the homotopy fiber of H(X,B^)H(X,B)\mathbf{H}(X,\hat B) \to \mathbf{H}(X,B) over cH(X,B)c \in \mathbf{H}(X,B) has, to the best of my (Urs Schreiber?) knowledge not been stated this way in the literature before. This arose in the course of the work

See there for examples and applications.

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

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

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

Revised on October 4, 2012 at 13:13:49 by Tim Porter