group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $f:\hat B \to B$ a morphism in an (∞,1)-topos? $\mathbf{H}$, twisted cohomology at stage $X$ is the fiber of $f$ over a given element? of $B$.
We now say this again, more formally and more in detail. For
$\mathbf{H}$ an (∞,1)-topos?;
$f:\hat B \to B$ a morphism in $\mathbf{H}$;
$X\in \mathbf{H}$ an object of $\mathbf{H}$;
and $c \in \mathbf{H}(X,B)$ a $B$-cocycle on $X$
twisted cohomology
is the set of equivalence classes of the fiber $\mathbf{H}^c(X,f)$ of the morphism $f:\hat B \to B$ over the generalized element? element $c$ of $B$. More explicitly, $\mathbf{H}^c(X,f)$ is the ∞-groupoid? defined as the homotopy pullback?
=–
The cocycle $c\in\mathbf{H}(X,B)$ induces the homotopy pullback
By the universal property of homotopy pullback, homotopy classes of sections $X \to P_f$, are in bijection with homotopy classes of homotopy commutative cones of the form
These in turn are manifestly the homotopy classes of maps $X \to \hat{B}$ such that $X \to \hat{B} \to B$ is homotopic to $c$. So
If $B$ is pointed, then the morphism $f:\hat{B}\to B$ induces a fibration sequence? $A \to \hat B \stackrel{f}{\to} B$ in $\mathbf{H}$, i.e. a homotopy pullback? square
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 $X$, there is fibration sequence of cocycle ∞-groupoids?
so that $\mathbf{H}^{const_{*}}(X,f)\simeq \mathbf{H}(X,A)$. In other words, twisted cohomology with the trivial twisting cocycle $const_{*}$ is nothing but cohomology with coefficients in the homotopy fiber of $f$ over the distinguished point of $B$. For this reason, when $B$ is pointed, it is customary to call the set of equivalence classes $\pi_0\mathbf{H}^c(X;f)$ the $c$-twisted $A$-cohomology of $X$, and to denote it by the symbol
The cohomology fibration sequence $\mathbf{H}(X,A) \to \mathbf{H}(X,\hat B) {\to} \mathbf{H}(X,B)$ can be seen as an obstruction problem in cohomology:
But it also says:
In the context of generalized (Eilenberg?Steenrod) cohomology? a coefficient object for cohomology? is a spectrum? $A$: the $A$-cohomology of a topological space? $X$ with coefficients in $A$ is the set of homotopy classes of maps $X \to A$. For instance, as a model of the degree-$0$ space in the K-theory spectrum? one can take $A = Fred(H)$, the space of Fredholm operators on a separable Hilbert space $H$. There is a canonical action on this space of the projective unitary group $G = P U(H)$ of $H$. Since $P U(H)$ has the homotopy type of an Eilenberg?Mac Lane space? $K(\mathbb{Z},2)$, a $P U(H)$-principal bundle? $P \to X$ defines a class $c \in H^3(X,\mathbb{Z})$ in ordinary integral cohomology? (this may also be thought of as the class of a twisting bundle gerbe?). The twisted K-theory (in degree $0$) of $X$ with that class as its twist is the set of homotopy classes of sections $X \to P \times_{P U(H)} Fred(H)$ of the associated bundle.
The above example generalizes straightforwardly to the case that
$A$ 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 $A$ can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces $\mathbf{B}G$ of nonabelian groups);
with a (topological) group? $G$ acting on $A$ by automorphisms and
a $G$-principal bundle? $P \to X.$
In this case there is an established definition of generalized (Eilenberg?Steenrod) cohomology? with coefficients $A$ twisted by a $G$-principal bundle? as follows.
From the $G$-principal bundle? $P \to X$ we obtain the associated $A$-bundle $P \times_G A \to X$. Then a twisted $A$-cocycle on $X$ is defined to be a section? $X \to P \times_G A$ of this associated bundle. The collection of homotopy classes of these sections is the twisted $A$-cohomology of $X$ with the twist specified by the class of $P$.
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 $G$ to be the trivial group and identify their $\Gamma$ and $\Pi$ with our $G$).
It is clearly a particular case of the general definition of twisted cohomology given above:
the $(\infty,1)$-topos $\mathbf{H}$ is the $(\infty,1)$-category of Top? of topological spaces
the object $B$ is the delooping? $\mathbf{B}G$ of the group? $G$.
the object $\hat{B}$ is the homotopy quotient $A//G\simeq \mathbf{E}G\times_G A$.
the twisting cocycle $c$ is the element in $\mathbf{Top}(X,\mathbf{B}G)$ defining the principal $G$-bundle $P\to X$.
Indeed, $B$ is pointed, we have a fibration sequence
and the homotopy pullback
is the $A$-bundle $P\times_G A\to X$.
The obstruction problem described by this example reads as folllows:
Read the other way round it says:
Since the associated bundle $P \times_G A$ 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 $P \times_G A$ as a parameterized family of spectra.
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.
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 $K \to \hat G \to G$ of ordinary groups as
classified by a group 2-cocycle? $\omega$, $c$-twisted $\hat G$-cohomology produces precisely the familiar notion of twisted bundles?, with the twist being the lifting gerbe? that obstructs the lift of a $G$-bundle to a \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.
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 $K(\pi,n)$, denoted $\tilde K(\pi,n)$, so that, for nice spaces, the cohomology with local coefficients $\tilde H^n(X,\pi)$ with respect to a homomorphism $\varepsilon:\pi_1(X)\to Aut(\pi)$ is given by homotopy classes of maps $X\to \tilde K(\pi,n)$ compatible with $\varepsilon.$
More generally, for any space $X$, let $A$ be a coefficient object that is equipped with an action of the first fundamental group? $\pi_1(X)$ of $X$. (Such an action is also called an $A$-valued local system? on $X$).
Then there is the fibration sequence?
of this action.
Notice that there is a canonical map $c : X \to \mathbf{B} \pi_1(X)$, the one that classifies the universal cover of $X$.
Then $A$-cohomology with local coefficients on $X$ is nothing but the $c$-twisted $A$-cohomology of $X$ in the above sense.
For the special case of generalized (Eilenberg?Steenrod) cohomology? twisted by a $G$-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 $c$-twisted cohomology as the homotopy fiber of $\mathbf{H}(X,\hat B) \to \mathbf{H}(X,B)$ over $c \in \mathbf{H}(X,B)$ 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.
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
Reidemeister (1938) (But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)
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 Farber, Katz, Levine Morse theory
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 $d+k\wedge$
2001 Barlewtta & Dragomir tc and integrability
2002 Lueck $L^2$
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