nLab
normal framing

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 $D- d$

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

(e.g. Kosinski 93, IX Theorem (5.5))

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.

Related concepts

• framed manifold

References

• Antoni Kosinski, chapter IX of Differential manifolds, Academic Press 1993 (pdf)