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 $K U$ 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 topological space $X$ is
The projective unitary group $P U(\mathcal{H})$ (a topological group) acts canonically by automorphisms on $(K U)_0$. 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
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
An original article is
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, Graeme Segal, Twisted K-theory (arXiv:math/0407054)
Michael Atiyah, Graeme Segal, Twisted K-theory and cohomology (arXiv:math/0510674)
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
Daniel Freed, Michael Hopkins, Constantin Teleman, Twisted K-theory and loop group representations (arXiv:math/0312155)
Daniel Freed, Michael Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory I (arXiv:0711.1906)
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 is discussed for instance in section 22 of
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).
Discussion in terms of vectorial bundles is in
The twisted version of differential K-theory is discussed in
Twists of $K \mathbb{R}$-theory relevant for orientifolds are discussed in