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

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

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 $D- d$

$$Cob_{Fr}^{d}(X) \underoverset{\simeq}{PT}{\longrightarrow} \pi^{D-d}(X) \,.$$

(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.

Related concepts

• framed link

• framed manifold

• normal twisted framing

References

• Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

• Lev Pontrjagin, Section 6 of: Smooth manifolds and their applications in homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955, AMS Translation Series 2, Vol. 11, 1959 (doi:10.1142/9789812772107_0001, pdf)

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