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 notion and term twisted bundle (with finite rank) apparently first appears in

The equivalent notion of gerbe module apparently appears first in

there explicitly in terms of Cech cocycles relative to an open cover. The generalization to infinite rank and arbitrary covering morphisms was amplified in (CBMMS) below.

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.

For twists $c$ that are torsion class (i.e. have finite order as group elements in the cohomology group $H(X,\mathbf{B}^2 A)$ ) this was realized in

which also, apparently, is the source where gerbe modules as such were first introduced.

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

There the twisted cocycle equation discussed above appears on the bottom of page 7.

Then there is

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

### 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