nLab
normal twisted framing

Contents

Context

Manifolds and cobordisms

Bundles

Contents

Definition

Definition

Consider a differentiable/smooth manifold M DM^D of dimension DD and equipped with a real vector bundle

(1)𝒱M dVectorBundles (M) \mathcal{V} \to M^d \;\;\;\; \in VectorBundles_{\mathbb{R}}(M)

of rank nDn \leq D, with classifying map to be denoted τ:MBO(n)\tau \;\colon\; M \longrightarrow B \mathrm{O}(n).

Then a τ\tau-twisted normal framing on a submanifold Σ dM\Sigma^d \hookrightarrow M is an isomorphism of real vector bundles

NΣ𝒱 |Σ N\Sigma \overset{\simeq}{\longrightarrow} \mathcal{V}_{\vert \Sigma}

between the normal bundle of the submanifold and the pullback bundle of 𝒱\mathcal{V} along its inclusion.

(e.g. Cruickshank 99, Def. 6.0.79) Cruickshank 03, Def. 5.1)

For this to exist it is necessary that

n=Dd n \;=\; D - d

is the codimension of Σ dM D\Sigma^d \hookrightarrow M^D.

Example

In the special case that 𝒱\mathcal{V} (1) is the trivial real vector bundle nn, the notion of twisted normal framing (Def. ) reduces to that of normal framing.

Properties

Twisted Pntrjagin theorem

The twisted Pontrjagin theorem identifies cobordism classes of twisted-framed submanifolds in M DM^D (Def. ) with the twisted Cohomotopy of M dM^d.

References

Created on March 3, 2021 at 08:43:06. See the history of this page for a list of all contributions to it.