Contents

cohomology

# Contents

## Idea

A concordance between cocycles in cohomology is a relation similar to but different from a plain coboundary, it is a “coboundary after geometric realization”.

A concordance is a left homotopy in an (∞,1)-topos with respect to a topological interval object, not with respect to the categorical interval .

For instance for $S = Diff$ the site of smooth manifolds, there is

• the “topological interval” $I \in \mathbf{H}_{diff}$ which is the smooth ∞-stack on $Diff$ represented by the manifold $I = [0,1]$;

• the “categorical interval” $Ex^\infty \Delta^1 \in \mathbf{H}_{Diff}$ is the smooth ∞-stack that is constant on the free groupoid on a single morphism.

## Definition

For $\mathbf{H}$ and (∞,1)-topos with a fixed notion of topological interval object $I$, for $A \in \mathbf{A}$ any coefficient object and $X \in \mathbf{H}$ any other object, a concordance between two objects

$c,d \in \mathbf{H}(X,A)$

(two cocycles in $A$-cohomology on $X$)

is an object $\eta \in A(X \times I)$ such that

$\begin{matrix} X&&\\ \downarrow&\searrow^{c}&\\ X \times I&\stackrel{\eta}{\to}& A\\ \uparrow& \nearrow_{d}&\\ X&& \end{matrix} \,.$

## Examples

### For topological principal bundles

###### Proposition

(concordant topological principal bundles are isomorphic)
For $X \,\in\, TopSp$ a topological space and $G \,\in\, Grp(TopSp)$ a topological group, consider a concordance between a pair of $G$-principal bundles over $X$,

If

or

(e.g. if $X$ admits the structure of a smooth manifold)

then there exists already an isomorphism of principal bundles

(e.g. Roberts & Stevenson 2016, Cor. 15)
###### Proof

Recall that isomorphisms between principal bundles $P$ and $P'$ over $X$ are equivalently global sections of the fiber bundle $(P \times_X P')/G$. In particular, for every $P$ the identity morphism on it corresponds to a canonical section of $(P \times_X P)/G$.

In the given situation, this means that we have a canonical local section $\sigma_0$ making the following solid diagram commute, exhibiting that the restriction of the bundle $P_0 \times [0,1]$ to $\{0\} \subset [0,1]$ is isomorphic to $P_0$, by construction:

Now

or

In either case, this implies that a lift exists, as shown by the dashed arrow above.

The resulting commutativity of the bottom right triangle says that this lift is a global section which hence exhibits an isomorphism of principal bundles (over $X \times [0,1]$) of this form:

$P \;\; \simeq_X \;\; P_0 \times [0,1] \,.$

The restriction of this isomorphism to $\{1\} \subset [0,1]$ is hence an isomorphism of the form $P_1 \,\simeq_X\, P_0$, as required.

### For topological vector bundles

For topological vector bundles over paracompact Hausdorff spaces, concordance classes coincide with plain isomorphism classes:

###### Proposition

(concordance of topological vector bundles)

Let $X$ be a paracompact Hausdorff space. If $E \to X \times [0,1]$ is a topological vector bundle over the product space of $X$ with the closed interval (hence a concordance of topological vector bundles on $X$), then the two endpoint-restrictions

$E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}}$

are isomorphic topological vector bundles over $X$.

For proof see at topological vector bundle this Prop..

### More examples

• For $A = VectrBund(-)$ the difference between concordance of vectorial bundles and isomorphism of vectorial bundles plays a crucial rule in the construction of K-theory from this model.

• The notions of coboundary and concordance exist in every cohesive (∞,1)-topos.

Discussion of concordance of topological principal bundles (in fact for simplicial principal bundles parameterized over some base space):

Discussion of concordance in terms of the shape modality in the cohesive (∞,1)-topos of smooth ∞-groupoids (see at shape via cohesive path ∞-groupoid for more):