group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Twisted K-theory is a twisted cohomology version of (topological) K-theory.
The most famous twist is by a class in degree 3 ordinary cohomology (geometrically a $U(1)$-bundle gerbe or circle 2-group-principal 2-bundle), but there are various other twists.
Write KU for the spectrum of complex topological K-theory. Its degree-0 space is, up to weak homotopy equivalence, the space
or the space $Fred(\mathcal{H})$ of Fredholm operators on some separable Hilbert space $\mathcal{H}$.
The ordinary topological K-theory of a suitable topological space $X$ is given by the set of homotopy classes of maps from (the suspension spectrum of) $X$ to $KU$:
The projective unitary group $P U(\mathcal{H})$ (a topological group) acts canonically by automorphisms on $(K U)_0$. (This follows by the identificatioon of $KU_0$ with the space of Fredholm operators, see below) Therefore for $P \to X$ any $PU(\mathcal{H})$-principal bundle, we can form the associated bundle $P \times_{P U(\mathcal{H})} (K U)_0$.
Since the homotopy type of $P U(\mathcal{H})$ is that of an Eilenberg-MacLane space $K(\mathbb{Z},2)$, there is precisely one isomorphism class of such bundles representing a class $\alpha \in H^3(X, \mathbb{Z})$.
The twisted K-theory with twist $\alpha \in H^3(X, \mathbb{Z})$ is the set of homotopy-classes of sections of such a bundle
Similarily the reduced $\alpha$-twisted K-theory is the subset
a
The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).
Write
$Cl_n \coloneqq Cl^{\mathbb{C}}(\mathbb{R}^n,\langle -,-\rangle)$
for the complexification of the Clifford algebra of the Cartesian space $\mathbb{R}^n$ with its standard inner product;
$S_n$ for its $\mathbb{Z}/2\mathbb{Z}$-graded irreducible module (see at spin representation);
$H_0$ for the $\mathbb{Z}/2\mathbb{Z}$-graded separable Hilbert space whose even and odd part are both infinite-dimensional.
For $n \in \mathbb{N}$, the topological space $Fred^{(n)}$ of Fredholm operators on $S_n \otimes H_0$ is the set
(where $\mathcal{B}$ denotes bounded operators and $\mathcal{K}$ denotes compact operators and where $[-,-]$ denotes the graded commutator) and the topology on this set is the subspace topology induced by the embedding
given by
where $\mathcal{B}$ is equipped with the compact-open topology and $\mathcal{K}$ with the norm topology.
(Atiyah-Singer 69, p. 7, Atiyah-Segal 04, p. 21, Freed-Hopkins-Teleman 11, def. A.40)
These spaces indeed form a model for the KU spectrum:
For all $n \in \mathbb{N}$ there are natural weak homotopy equivalences
and
between the spaces of graded Fredholm operators of def. 2 and their loop spaces.
(Atiyah-Singer 69, theorem B(k), Atiyah-Segal 04 (4.2), Freed-Hopkins-Teleman 11, below def. A.40)
Regard the stable unitary group $U(H_0)$ as equipped with the subspace topology induced by the inclusion
from the compact-open topology on the bounded linear operators.
The conjugation action of the stable unitary group $U(H_0)$ on $Fred^{(n)}$, def. 2, is continuous.
This follows with (Atiyah-Segal 04, prop. A1.1).
Given a class $\chi \in H^3(X,\mathbb{Z})$ represented by a $PU(H_0)$-bundle $P \to X$ with associated Fredholm bundle
then the corresponding $\chi$-twisted cohomology spectrum is that consisting of the spaces of sections
(Freed-Hopkins-Teleman 11, def. 3.14)
Let $\alpha \in H^3(X, \mathbb{Z})$ be a class in degree-3 integral cohomology and let $P \in \mathbf{H}^3(X, \mathbf{B}^2 U(1))$ be any cocycle representative, which we may think of either as giving a circle 2-bundle or a bundle gerbe.
Write $TwBund(X, P)$ for the groupoid of twisted bundles on $X$ with twist given by $P$. Then let
be the set of isomorphism classes of twisted bundles. Call this the twisted K-theory of $X$ with twist $\alpha$.
(Some technical details need to be added for the non-torsion case.)
This definition of twisted $K_0$ is equivalent to that of prop. 1.
This is (CBMMS, prop. 6.4, prop. 7.3).
A circle 2-group principal 2-bundle is also incarnated as a centrally extended Lie groupoid. The corresponding twisted groupoid convolution algebra has as its operator K-theory the twisted K-theory of the base space (or base-stack). See at KK-theory for more on this.
Let $Vectr$ be the stack of vectorial bundles. (If we just take vector bundles we get a notion of twisted K-theory that only allows twists that are finite order elements in their cohomology group).
There is a canonical morphism
coming from the standard representation of the group $U(1)$.
Let $\mathbf{B}_{\otimes} Vectr$ be the delooping of $Vectr$ with respect to the tensor product monoidal structure (not the additive structure).
Then we have a fibration sequence
of (infinity,1)-categories (instead of infinity-groupoids).
The entire morphism above deloops
being the standard representation of the 2-group $\mathbf{B}U(1)$.
From the general nonsense of twisted cohomology this induces canonically now for every $\mathbf{B}^2 U(1)$-cocycle $c$ (for instance given by a bundle gerbe) a notion of $c$-twisted $Vectr$-cohomology:
After unwrapping what this means, the result of (Gomi) shows that concordance classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory.
By the general discussion of twisted cohomology the moduli space for the twists of periodic complex K-theory $KU$ is the Picard ∞-group in $KU Mod$. The “geometric” twists among these have as moduli space the non-connected delooping $bgl_1^\ast(KU)$ of the ∞-group of units of $KU$.
A model for this in 4-truncation is given by super line 2-bundles. For the moment see there for further discussion and further references.
twisted K-theory
A textbook account is in
The concept of twisted K-theory originates in
Max Karoubi, Algèbres de Clifford et K-théorie. Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968).
Peter Donovan, Max Karoubi, Graded Brauer groups and $K$-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)
which discusses twists of $KO$ and $KU$ over some $X$ by elements in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$.
The formulation in terms of sections of Fredholm bundles seems to go back to
A comprehensive account of twisted K-theory with twists in $H^3(X, \mathbb{Z})$ is in
Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’Institut des Hautes Études Scientifiques January 1969, Volume 37, Issue 1, pp 5-26 (pdf)
Michael Atiyah, Graeme Segal, Twisted K-theory (arXiv:math/0407054)
Michael Atiyah, Graeme Segal, Twisted K-theory and cohomology (arXiv:math/0510674)
and for more general twists in
The seminal result on the relation to loop group representations, now again with twists in $H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})$, is in the series of articles
The result on twisted K-groups has been lifted to an equivalence of categories in
Discussion in terms of Karoubi K-theory/Clifford module bundles is in
The perspective of twisted K-theory by sections of a $K U$-bundle of spectra (parameterized spectra) is discussed in
May, Sigurdsson, section 22 of Parametrized homotopy theory (pdf) AMS Lecture notes 132
Matthew Ando, Andrew Blumberg, David Gepner, section 7 of Twists of K-theory and TMF, in Jonathan Rosenberg et al. (eds.), Superstrings, Geometry, Topology, and $C^\ast$-algebras, volume 81 of Proceedings of Symposia in Pure Mathematics, 2009 (arXiv:1002.3004)
See the references at (infinity,1)-vector bundle for more on this.
Discussion in terms of twisted bundles/bundle gerbe modules is in
and for generalization to groupoid K-theory also (FHT 07, around p. 26) and
(which establishes the relation to KK-theory).
Max Karoubi, Twisted bundles and twisted K-theory, arxiv/1012.2512
Ulrich Pennig, Twisted K-theory with coefficients in $C^\ast$-algebras, (arXiv:1103.4096)
Discussion in terms of vectorial bundles is in
Kiyonori Gomi, Twisted K-theory and finite-dimensional approximation (arXiv)
Kiyonori Gomi und Yuji Terashima, Chern-Weil Construction for Twisted K-Theory Communication ins Mathematical Physics, Volume 299, Number 1, 225-254,
The twisted version of differential K-theory is discussed in
Twists of $K \mathbb{R}$-theory relevant for orientifolds are discussed in
Discussion of twisted K-homology:
Bai-Ling Wang, Gemometric cycles, index theory and twisted K-homology (arXiv:0710.1625)
Eckhard Meinrenken, Twisted K-homology and group-valued moment maps, International Mathematics Research Notices 2012 (20) (2012), 4563–4618 (arXiv:1008.1261)
Bei Liu, Twisted K-homology,Geometric cycles and T-duality (arXiv:1411.1575)
Discussion of combined twisted and equivariant and real K-theory