nLab
normal framing

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Context

Manifolds and cobordisms

Bundles

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 (Pontrjagin 55, Sec. 6 e.g. Kosinski 93, IX (2.1)). Beware that this is often called just a framed submanifold, despite the potential class with “framed manifold”.

Properties

Pontryagin’s theorem

For XX a closed smooth manifold of dimension DD, the Pontryagin theorem (e.g. Kosinski 93, IX.5) identifies the set

Cob Fr d(X) Cob_{Fr}^{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

Cob Fr d(X)PTπ Dd(X). Cob_{Fr}^{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 March 3, 2021 at 08:42:51. See the history of this page for a list of all contributions to it.