manifolds and cobordisms
cobordism theory, Introduction
A normal framing is a trivialization of a normal bundle.
Specifically, if is a smooth manifold and is a submanifold, then a normal framing for is a trivialization of the normal bundle .
A submanifold equipped with such normal framing is a normally framed submanifold (e.g. Kosinski 93, IX (2.1)).
For a closed smooth manifold of dimension , the Pontryagin-Thom construction (e.g. Kosinski 93, IX.5) identifies the set
of cobordism classes of closed and normally framed submanifolds of dimension inside with the cohomotopy of in degree
(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.
Last revised on February 5, 2019 at 06:36:07. See the history of this page for a list of all contributions to it.