Contents

# Contents

## Definition

A normal framing is a trivialization of a normal bundle.

Specifically, if $X$ is a smooth manifold and $\Sigma \overset{\iota}{\hookrightarrow} X$ is a submanifold, then a normal framing for $\Sigma$ is a trivialization of the normal bundle $N_\iota(X)$.

A submanifold equipped with such normal framing is a normally framed submanifold (e.g. Kosinski 93, IX (2.1)).

## Properties

### Pontryagin-Thom isomorphism

For $X$ a closed smooth manifold of dimension $D$, the Pontryagin-Thom construction (e.g. Kosinski 93, IX.5) identifies the set

$SubMfd_{/bord}^{d}(X)$

of cobordism classes of closed and normally framed submanifolds $\Sigma \overset{\iota}{\hookrightarrow} X$ of dimension $d$ inside $X$ with the cohomotopy $\pi^{D-d}(X)$ of $X$ in degree $D- d$

$SubMfd_{/bord}^{d}(X) \underoverset{\simeq}{PT}{\longrightarrow} \pi^{D-d}(X) \,.$

In particular, by this bijection the canonical group structure on cobordism groups in sufficiently high codimension (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets in sufficiently high degree.

## References

Last revised on February 5, 2019 at 06:36:07. See the history of this page for a list of all contributions to it.