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 $U\left(1\right)$-bundle gerbe or circle 2-group-principal 2-bundle), but there are various other twists.

## Definition

### By sections of associated $KU$-bundles

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

$BU×ℤ={\underset{\to }{\mathrm{lim}}}_{n}BU\left(n\right)×ℤ$B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z}

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

$\left(KU{\right)}_{0}\simeq BU×ℤ\simeq \mathrm{Fred}\left(ℋ\right)\phantom{\rule{thinmathspace}{0ex}}.$(K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,.

The ordinary topological K-theory of a topological space $X$ is

$K\left(X{\right)}_{•}\simeq \left[X,\left(KU{\right)}_{•}\right]\phantom{\rule{thinmathspace}{0ex}}.$K(X)_\bullet \simeq [X, (K U)_\bullet] \,.

The projective unitary group $PU\left(ℋ\right)$ (a topological group) acts canonically by automorphisms on $\left(KU{\right)}_{0}$. Therefore for $P\to X$ any $\mathrm{PU}\left(ℋ\right)$-principal bundle, we can form the associated bundle $P{×}_{PU\left(ℋ\right)}\left(KU{\right)}_{0}$.

Since the homotopy type of $PU\left(ℋ\right)$ is that of an Eilenberg-MacLane space $K\left(ℤ,2\right)$, there is precisely one isomorphism class of such bundles representing a class $\alpha \in {H}^{3}\left(X,ℤ\right)$.

###### Definition

The twisted K-theory with twist $\alpha \in {H}^{3}\left(X,ℤ\right)$ is the set of homotopy-classes of sections of such a bundle

${K}_{\alpha }\left(X{\right)}^{0}:={\Gamma }_{X}\left({P}^{\alpha }{×}_{PU\left(ℋ\right)}\left(KU{\right)}_{0}\right)\phantom{\rule{thinmathspace}{0ex}}.$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

${\stackrel{˜}{K}}_{\alpha }\left(X{\right)}^{0}:={\Gamma }_{X}\left({P}^{\alpha }{×}_{PU\left(ℋ\right)}BU\right)\phantom{\rule{thinmathspace}{0ex}}.$\tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,.

### By twisted vector bundles (gerbe modules)

###### Definition

Let $\alpha \in {H}^{3}\left(X,ℤ\right)$ be a class in degree-3 integral cohomology and let $P\in {H}^{3}\left(X,{B}^{2}U\left(1\right)\right)$ be any cocycle representative, which we may think of either as giving a circle 2-bundle or a bundle gerbe.

Write $\mathrm{TwBund}\left(X,P\right)$ for the groupoid of twisted bundles on $X$ with twist given by $P$. Then let

${\stackrel{˜}{K}}_{\alpha }\left(X\right):=\mathrm{TwBund}\left(X,P\right)$\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. 1.

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 $\mathrm{Vectr}$ 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

$\rho :BU\left(1\right)\to \mathrm{Vect}↪\mathrm{Vectr}$\rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr

coming from the standard representation of the group $U\left(1\right)$.

Let ${B}_{\otimes }\mathrm{Vectr}$ be the delooping of $\mathrm{Vectr}$ with respect to the tensor product monoidal structure (not the additive structure).

Then we have a fibration sequence

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

The entire morphism above deloops

$B\rho :{B}^{2}U\left(1\right)\to {B}_{\otimes }\mathrm{Vect}↪{B}_{\otimes }\mathrm{Vectr}$\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 $BU\left(1\right)$.

From the general nonsense of twisted cohomology this induces canonically now for every ${B}^{2}U\left(1\right)$-cocycle $c$ (for instance given by a bundle gerbe) a notion of $c$-twisted $\mathrm{Vectr}$-cohomology:

$\begin{array}{ccc}{H}^{c}\left(X,\mathrm{Vectr}\right)& \to & *\\ ↓& & {↓}^{B\rho \circ c}\\ *& \to & H\left(X,{B}_{\otimes }\mathrm{Vectr}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${H}^{c}\left(X,\mathrm{Vectr}\right)$ yield twisted K-theory.

## References

An original article is

• 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 $\mathrm{KO}$ and $\mathrm{KU}$ over some $X$ by elements in ${H}^{0}\left(X,{ℤ}_{2}\right)×{H}^{1}\left(X,{ℤ}_{2}\right)×{H}^{3}\left(X,ℤ\right)$.

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 ${H}^{3}\left(X,ℤ\right)$ is in

The seminal result on the relation to loop group representations, now again with twists in ${H}^{0}\left(X,{ℤ}_{2}\right)×{H}^{1}\left(X,{ℤ}_{2}\right)×{H}^{3}\left(X,ℤ\right)$, is in the series of articles

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 $KU$-bundle of spectra is discussed for instance in section 22 of

• May, Sigurdsson, Parametrized homotopy theory (pdf) AMS Lecture notes 132

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

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

Discussion in terms of vectorial bundles is in

• 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 $Kℝ$-theory relevant for orientifolds are discussed in

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

Revised on April 4, 2013 00:48:22 by Urs Schreiber (82.169.65.155)