nLab
concordance

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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=DiffS = Diff the site of smooth manifolds, there is

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

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

Definition

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

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

(two cocycles in AA-cohomology on XX)

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

X c X×I η A d X . \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 XTopSpX \,\in\, TopSp a topological space and GGrp(TopSp)G \,\in\, Grp(TopSp) a topological group, consider a concordance between a pair of GG-principal bundles over XX,

If

or

(e.g. if XX 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 PP and PP' over XX are equivalently global sections of the fiber bundle (P× XP)/G(P \times_X P')/G. In particular, for every PP the identity morphism on it corresponds to a canonical section of (P× XP)/G(P \times_X P)/G.

In the given situation, this means that we have a canonical local section σ 0\sigma_0 making the following solid diagram commute, exhibiting that the restriction of the bundle P 0×[0,1]P_0 \times [0,1] to {0}[0,1]\{0\} \subset [0,1] is isomorphic to P 0P_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×[0,1]X \times [0,1]) of this form:

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

The restriction of this isomorphism to {1}[0,1]\{1\} \subset [0,1] is hence an isomorphism of the form P 1 XP 0P_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 XX be a paracompact Hausdorff space. If EX×[0,1]E \to X \times [0,1] is a topological vector bundle over the product space of XX with the closed interval (hence a concordance of topological vector bundles on XX), then the two endpoint-restrictions

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

are isomorphic topological vector bundles over XX.

For proof see at topological vector bundle this Prop..

More examples

  • For A=VectrBund()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):

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):

Last revised on October 21, 2021 at 11:22:56. See the history of this page for a list of all contributions to it.