Contents

cohomology

# Contents

## Idea

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.

## Definition

### By sections of associated $K U$-bundles

Write KU for the spectrum of complex topological K-theory. Its degree-0 space is, up to weak homotopy equivalence, the space

$B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z}$

or the space $Fred(\mathcal{H})$ of Fredholm operators on some separable Hilbert space $\mathcal{H}$.

$(K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,.$

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 $KU$:

$K(X)_\bullet \simeq [X, (K U)_\bullet] \,.$

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 $\alpha \in H^3(X, \mathbb{Z})$.

###### Definition

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

$K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0) \,.$

Similarily the reduced $\alpha$-twisted K-theory is the subset

$\tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,.$

a

### By bundles of Fredholm operators

The following is due to (Atiyah-Singer 69, Atiyah-Segal 04).

Write

###### Definition

For $n \in \mathbb{N}$, the topological space $Fred^{(n)}$ of Fredholm operators on $S_n \otimes H_0$ is the set

$Fred^{(n)} \coloneqq \left\{ F \in \mathcal{B}(S_n \otimes H_0) \;|\; F \, odd\,, F^\ast = F \,, F^2 - 1 \in \mathcal{K}(S_n \otimes H_0)\,, [F,\gamma] = 0 \, for\, \gamma \in Cl_n \right\}$

(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

$Fred^{(n)} \hookrightarrow \mathcal{B}(S_n \otimes H_0) \times \mathcal{K}(S_n \otimes H_0)$

given by

$F\mapsto (F, F^2 - 1) \,,$

where $\mathcal{B}$ is equipped with the compact-open topology and $\mathcal{K}$ with the norm topology.

These spaces indeed form a model for the KU spectrum:

###### Proposition

For all $n \in \mathbb{N}$ there are natural weak homotopy equivalences

$Fred^{(n+1)} \stackrel{\simeq}{\longrightarrow} \Omega Fred^{(n)}$

and

$Fred^{(n+2)} \stackrel{\simeq}{\longrightarrow} Fred^{(n)}$

between the spaces of graded Fredholm operators of def. and their loop spaces.

Regard the stable unitary group $U(H_0)$ as equipped with the subspace topology induced by the inclusion

$U(H_0) \stackrel{(id,(-)^{-1})}{\hookrightarrow} \mathcal{B}(H_0)\times\mathcal{B}(H_0)$

from the compact-open topology on the bounded linear operators.

###### Proposition

The conjugation action of the stable unitary group $U(H_0)$ on $Fred^{(n)}$, def. , is continuous.

This follows with (Atiyah-Segal 04, prop. A1.1).

###### Definition

Given a class $\chi \in H^3(X,\mathbb{Z})$ represented by a $PU(H_0)$-bundle $P \to X$ with associated Fredholm bundle

$Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)} \,,$

then the corresponding $\chi$-twisted cohomology spectrum is that consisting of the spaces of sections

$\Gamma(X, Fred^{(n)+ \chi}) \,.$

### By twisted vector bundles (gerbe modules)

###### Definition

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

$\tilde K_\alpha(X) := TwBund(X,P)$

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.)

###### Proposition

This definition of twisted $K_0$ is equivalent to that of prop. .

This is (CBMMS, prop. 6.4, prop. 7.3).

### By KK-theory of twisted convolution algebras

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.

## Other constructions

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

$\rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr$

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

$Vectr \to {*} \to \mathbf{B}_\otimes Vectr$

The entire morphism above deloops

$\mathbf{B}\rho : \mathbf{B}^2 U(1) \to \mathbf{B}_\otimes Vect \hookrightarrow \mathbf{B}_{\otimes} Vectr$

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:

$\array{ \mathbf{H}^c(X, Vectr) &\to& {*} \\ \downarrow && \downarrow^{\mathbf{B}\rho \circ c} \\ {*} &\to& \mathbf{H}(X,\mathbf{B}_\otimes Vectr) } \,.$

After unwrapping what this means, the result of (Gomi) shows that concordance classes in $\mathbf{H}^c(X,Vectr)$ yield twisted K-theory.

## Twists

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.

## References

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

• Jonathan Rosenberg, Continuous-trace algebras from the bundle theoretic point of view , J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381.

A comprehensive account of twisted K-theory with twists in $H^3(X, \mathbb{Z})$ is in

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

• Max Karoubi, Clifford modules and twisted K-theory, Proceedings of the International Conference on Clifford algebras (ICCA7) (arXiv:0801.2794)

The perspective of twisted K-theory by sections of a $K U$-bundle of spectra (parameterized spectra) is discussed in

See the references at (infinity,1)-vector bundle for more on this.

Discussion in terms of twisted bundles/bundle gerbe modules is in

but apparently contains a mistake, as pointed out in

The generalization of this to groupoid K-theory is in (FHT 07, around p. 26) and

(which establishes the relation to KK-theory).

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

• Alan Carey, Differential twisted K-theory and applications ESI preprint (pdf)

Twists of $K \mathbb{R}$-theory relevant for orientifolds are discussed in

• El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)

Discussion of twisted K-homology:

Discussion of combined twisted and equivariant and real K-theory

Last revised on September 6, 2018 at 03:30:04. See the history of this page for a list of all contributions to it.