nLab concordance

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)
With $kTopSp$ denoting the category of compactly generated weakly Hausdorff spaces, for $X \,\in\, kTopSp$ a k-topological space and $G \,\in\, Grp(kTopSp)$ a $k$-topological group, consider a concordance between a pair of $G$-principal bundles over $X$,

If

• $X$ is a CW-complex, or, more generally, a retract of a cell complex

or

• $P$ is numerable, e.g. in that $X$ is a paracompact topological space

(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

Observe that isomorphisms $f \,\colon\, P \xrightarrow{\;} P'$ between principal bundles over $X$ are equivalently global sections of the fiber bundle $(P \times_X P')/G$:

Here, from left to right, the dashed section follows by the universal property of the quotient space $X = P/G$. From right to left, the top morphism follows by pullback along the dashed section, using that

1. the bundle projections are effective epimorphisms by local triviality,

2. their kernel pairs are as shown, by principality,

3. compactly generated topological spaces form a regular category (by this Prop.),

4. in a regular category pullback preserves effective epimorphisms (this Prop.) and, of course, their kernel pairs.

In particular, for every $P$ the identity morphism on it corresponds to the 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

• assuming the first condition:

  1. the right vertical map is a Serre fibration, as all locally trivial fiber bundles are Serre fibrations (by this Prop.);

  2. the left vertical map is a Serre-Quillen-acyclic cofibration – since (see this Prop.) it is the product $id_X \times (D^0 \hookrightarrow D^0 \times [0,1] )$ of the cofibrant object $X$ with a generating acyclic cofibration (see this Def.) –, hence is a Serre-Quillen-acyclic cofibration and as such has the left lifting property against the right map;

or

• assuming the second condition:

  1. the right vertical map is a Hurewicz fibration, by this Prop.,

  2. hence it has the right lifting property against the left map, by definition of Hurewicz fibrations.

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.

References

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

• David Roberts, Danny Stevenson, Cor. 15 in: Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)

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):

• Dmitri Pavlov, Structured Brown representability via concordance, 2014 (pdf, pdf)

• Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves (arXiv:1912.10544)