Contents

cohomology

# Contents

## Idea

A twisted principal-bundle is the object classified by a cocycle in twisted cohomology the way an ordinary principal bundle is the object classified by a cocycle in plain cohomology (generally in nonabelian cohomology).

For $\hat G$ a group, a $\hat G$-principal bundle is classified in degree 1 nonabelian cohomology with coefficients in the delooped groupoid $\mathbf{B} \hat G$.

Given a realization of $\hat G$ as an abelian extension

$A \to \hat G \to G$

of groups, i.e. given a fibration sequence

$\mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G$

of groupoids such that $\mathbf{B}A$ is once deloopable so that the fibration sequence continues to the right at least one step as

$\mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A$

the general mechanism of twisted cohomology induces a notion of twisted $\hat G$-cohomology. The fibrations classified by this are the twisted $\hat G$-bundles.

## Definition

We give a discussion of twisted bundles as a realization of twisted cohomology in any cohesive (∞,1)-topos $\mathbf{H}$ as described in the section cohesive (∞,1)-topos – twisted cohomology. For the cases that $\mathbf{H} =$ ETop∞Grpd or $\mathbf{H} =$ Smooth∞Grpd this reproduces the traditional notion of topological and smooth twisted bundles, respectively, whose twists are correspondingly topological or smooth bundle gerbes/circle n-bundles.

### Setup

Let $\mathbf{B}^{n-1}U(1) \in \mathbf{H}$ be the circle n-group. We shall concentrate here for definiteness on twists in $\mathbf{B}^2 U(1)$-cohomology, since that reproduces the usual notions of twisted bundles found in the literature. But every other choice would work, too, and yield a corresponding notion of twisted bundles.

Fix once and for all an ∞-group $G \in \mathbf{H}$ and a cocycle

$\mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 U(1)$

representing a characteristic class

$[\mathbf{c}] \in H_{Smooth}^2(\mathbf{B}G,U(1))$

Notice that if $G$ is a compact Lie group, as usual for the discussion of twisted bundles where $G = P U(n)$ is the projective unitary group in some dimension $n$, then by this theorem we have that

$H_{Smooth}^2(\mathbf{B}G, U(1)) \simeq H^3(B G, \mathbb{Z}) \,,$

where on the right we have the ordinary integral cohomology of the classifying space $B G \in$ Top of $G$.

### The abstract definition

Let $G$ and $\mathbf{c}$ be as above.

###### Definition

Write

$\mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 U(1)$

for the homotopy fiber of $\mathbf{c}$.

This identifies $\hat G$ as the group extension of $G$ by the 2-cocycle $\mathbf{c}$.

###### Note

Equivalently this means that

$\mathbf{B}U(1) \to \mathbf{B}\hat G \to \mathbf{B}G$

is the smooth circle 2-bundle/bundle gerbe classified by $\mathbf{c}$; and its loop space object

$U(1) \to \hat G \to G$

the corresponding circle group principal bundle on $G$.

Let $X \in \mathbf{H}$ be any object. From twisted cohomology we have the following notion.

###### Definition

The degree-1 total twisted cohomology $H_{tw}^1(X, \hat G)$ of $X$ with coefficients in $\hat G$, def. , relative to the characteristic class $[\mathbf{c}]$ is the set

$H^1_{tw}(X, \hat G) := \pi_0 \mathbf{H}_{tw}(X, \mathbf{G}\hat H)$

of connected components of the (∞,1)-pullback

$\array{ \mathbf{H}_{tw}(X, \mathbf{B}\hat G) &\stackrel{tw}{\to}& H_{Smooth}^2(X,U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{c}_*}{\to}& \mathbf{H}(X, \mathbf{B}^2 U(1)) } \,,$

where the right verticsl morphism is any section of the truncation projection from cocycles to cohomology classes.

Given a twisting class $[\alpha] \in H^2_{Smooth}(U(1))$ we say that

$H_{[\alpha]}^1(X,\hat G) := H^1_{tw}(X, \hat G) \times_{[\alpha]} *$

is the $[\alpha]$-twisted cohomology of $X$ with coefficients in $\hat G$ relative to $\mathbf{c}$.

###### Note

For $[\alpha] = 0$ the trivial twist, $[\alpha]$-twisted cohomology coincides with ordinary cohomology:

$H^1_{[\alpha] = 0}(X, \hat G) \simeq H^1_{Smooth}(X, \hat G) \,.$

By the discussion at principal ∞-bundle we may identify the elements of $H^1_{Smooth}(X, \hat G)$ with $\hat G$-principal ∞-bundles $P \to X$. In particular if $\hat G$ is an ordinary Lie group and $X$ is an ordinary smooth manifold, then these are ordinary $\hat G$-principal bundles over $X$. This justifies equivalently calling the elements of $H^1_{tw}(X,\hat G)$ twisted principal $\infty$-bundles; and we shall write

$\hat G TwBund(X) := H^1_{tw}(X, \hat G) \,,$

where throughout we leave the characteristic class $[\mathbf{c}]$ with respect to which the twisting is defined implcitly understood.

### Explicit cocycles

We unwind the abstract definition, def. , to obtain the explicit definition of twisted bundles by Cech cocycles the way they appear in the traditional literature (see the General References below).

###### Proposition

Let $U(1) \to \hat G \to G$ be a group extension of topological groups.

Let $X \in$ Mfd $\hookrightarrow$ ETop∞Grpd $=: \mathbf{H}$ be a paracompact topological manifold with good open cover $\{U_i \to X\}$.

1. Relative to this every twisting cocycle $[\alpha] \in H^2_{ETop}(X, U(1))$ is a Cech cohomology representative given by a collection of functions

$\{ \alpha_{i j k} : U_i \cap U_j \cap U_k \to U(1) \}$

satisfying on every quadruple intersection the equation

$\alpha_{i j k} \alpha_{i k l} = \alpha_{j k l} \alpha_{i j l} \,.$
2. I terms of this cocycle data the twisted cohomology $H^1_{[\alpha]}(X, \hat G)$ is given by equivalence classes of cocycles consisting of

1. collections of functions

$\{g_{i j} : U_i \cap U_j \to \hat G \}$

subject to the condition that on each triple overlap the equation

$g_{i j} \dot g_{j k} = g_{i k} \cdot \alpha_{i j k}$

holds, where on the right we are injecting $\alpha_{i j k}$ via $U(1) \to \hat G$ into $\hat G$
and then form the product there;

2. subject to the equivalence relation that identifies two such collections of cocycle data $\{g_{i j}\}$ and $\{g'_{i j}\}$ if there exists functions

$\{h_i : U_i \to \hat G\}$

and

$\{\beta_{i j} : U_i \cap U_j \to \hat U(1)\}$

such that

$\beta_{i j} \beta_{j k} = \beta_{i k}$

and

$g'_{i j} = h_i^{-1} \cdot g_{i j} \cdot h_j \cdot \beta_{i j} \,.$
###### Proof

We pass to the standard presentation of ETop∞Grpd by the projective local model structure on simplicial presheaves over the site CartSp. We then compute the defining (∞,1)-pullback by a homotopy pullback there.

Write $\mathbf{B}\hat G_{c}, \mathbf{B}^2 U(1)_c \in [CartSp^{op}, sSet]$ etc. for the standard models of the abstract objects of these names by simplicial presheaves. Write accordingly $\mathbf{B}(U(1) \to \hat G)_c$ for the delooping of the crossed module associated to the central extension $\hat G \to G$.

In terms of this the characteristic class $\mathbf{c}$ is represented by the ∞-anafunctor

$\array{ \mathbf{B}(U(1) \to \hat G)_c &\stackrel{\mathbf{c}}{\to}& \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G_c } \,,$

where the top horizontal morphism is the evident projection onto the $U(1)$-labels. Moreover, the Cech nerve of the good open cover $\{U_i \to X\}$ forms a cofibrant resolution

$\emptyset \hookrightarrow C(\{U_i\}) \stackrel{\simeq}{\to} X$

and so $\alpha$ is presented by an ∞-anafunctor

$\array{ C(\{U_i\}) &\stackrel{\alpha}{\to}& \mathbf{B}^2 U(1)_c \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.$

Using that $[CartSp^{op}, sSet]_{proj}$ is a simplicial model category this means in conclusion that the homotopy pullback in question is given by the ordinary pullback of simplicial sets

$\array{ \mathbf{H}^1_{[\alpha]}(X,\hat G) &\to& * \\ \downarrow && \downarrow^{\mathrlap{\alpha}} \\ [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}(U(1) \to \hat G)_c) &\stackrel{\mathbf{c}_*}{\to}& [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}^2 U(1)_c) } \,.$

An object of the resulting simplicial set is then seen to be a simplicial map $g : C(\{U_i\}) \to \mathbf{B}(U(1) \to \hat G)_c$ that assigns

$g \;\; : \;\; \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\;\; \mapsto \;\;\;\; \array{ && \bullet \\ & {}^{\mathllap{g_{i j}(x)}}\nearrow &\Downarrow^{\alpha_{i j k}(x)}& \searrow^{\mathrlap{g_{j k}(x)}} \\ \bullet &&\underset{g_{i k}(x)}{\to}&& \bullet }$

such that projection out along $\mathbf{B}(U(1) \to \hat G)_c \to \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c$ produces $\alpha$.

Similarily for the morphisms. Writing out what these diagrams in $\mathbf{B}(U(1) \to \hat G)_c$ mean in equations, one finds the formulas claimed above.

## Properties

### General

(…)

Consider the extension $U(1) \to U(n) \to P U(n)$ of the projective unitary group to the unitary group for all $n$. Then direct sum of matrices gives a sum operation

$H^1_{[\alpha]}(X, P U(n_1)) \times H^1_{[\alpha]}(X, P U(n_2)) \to H^1_{[\alpha]}(X, P U(n_1 + n_2))$

and a tensor product operation

$H^1_{[\alpha_1]}(X, P U(n)) \times H^1_{[\alpha_2]}(X, P U(n)) \to H^1_{[\alpha_1]+ [\alpha_2]}(X, P U(n_1 \cdot n_2))$

(…)

### Twisted K-theory

Equivalence classes of twisted $U(n)$-bundles for fixed $\mathbf{B}U(1)$-twist $\alpha$ form a model for topological $\alpha$-twisted K-theory. See there for details.

### General

The concept of twisted vector bundles was introduced, as a model for twisted K-theory (in the generality of equivariant bundles and orbifold K-theory) in:

The Cech cocycle-incarnation of twisted vector bundles (effectively due to Lupercio & Uribe 2001, Def. 7.2.1) was considered in:

An equivalent characterization of twisted vector bundles (identified in Lupercio & Uribe 2001, (v2-) Prop. 7.2.2) appeared under the name bundle gerbe module in:

Discussion of a splitting principle for twisted vector bundles (phrased in terms of gerbe modules) is in

• Atsushi Tomoda, On the splitting principle of bundle gerbe modules, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (Euclid, talk slides pdf)

### In twisted K-theory

Just as vector bundles model cocycles in K-theory, twisted vector bundles model cocycles in twisted K-theory.

The generalization of this construction to non-torsion twists requires using vectorial bundles instead of plain vector bundles. Full twisted K-theory in terms of twisted vectorial bundles was realized in

Then there is

• Max Karoubi, Twisted bundles and twisted K-theory, Clay Mathematics Proceedings, Volume 19 (2011) (pdf)

• Ulrich Pennig, Twisted K-theory with coefficients in $C^\ast$-algebras, (arXiv:1103.4096)

### As 2-sections of 2-bundles

The observation that twisted vector bundles may be understood as higher-order sections of 2-vector bundles associated with circle 2-bundles/bundle gerbes appears in

• Urs Schreiber, Quantum 2-States: Sections of 2-vector bundles Talk at Higher categories and their applications, Fields institute (2007) (pdf).

A discussion of this with 2-connections taken into account is in section 4.4.3 of

A discussion in the context of principal infinity-bundles (as opposed to higher vector bundles), is in section “2.3.5 Twisted cohomology and sections” and then in section “3.3.7.2 Twisted 1-bundles – twisted K-theory”

The observation then re-appears independently in