nLab
normal framing

Contents

Contents

Definition

A normal framing is a trivialization of a normal bundle.

Specifically, if XX is a smooth manifold and ΣιX\Sigma \overset{\iota}{\hookrightarrow} X is a submanifold, then a normal framing for Σ\Sigma is a trivialization of the normal bundle N ι(X)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 XX a closed smooth manifold of dimension DD, the Pontryagin-Thom construction (e.g. Kosinski 93, IX.5) identifies the set

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

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

SubMfd /bord d(X)PTπ Dd(X). 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.

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.