# nLab normal twisted framing

Contents

### Context

#### Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

bundles

# Contents

## Definition

###### Definition

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

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

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

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

$N\Sigma \overset{\simeq}{\longrightarrow} \mathcal{V}_{\vert \Sigma}$

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

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

For this to exist it is necessary that

$n \;=\; D - d$

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

###### Example

In the special case that $\mathcal{V}$ (1) is the trivial real vector bundle $n$, 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 $M^D$ (Def. ) with the twisted Cohomotopy of $M^d$.

## References

Created on March 3, 2021 at 13:43:06. See the history of this page for a list of all contributions to it.