normal framing

**manifolds** and **cobordisms**

cobordism theory, *Introduction*

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)).

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.

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

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