manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
Consider a differentiable/smooth manifold of dimension and equipped with a real vector bundle
of rank , with classifying map to be denoted .
Then a -twisted normal framing on a submanifold is an isomorphism of real vector bundles
between the normal bundle of the submanifold and the pullback bundle of along its inclusion.
(e.g. Cruickshank 99, Def. 6.0.79) Cruickshank 03, Def. 5.1)
For this to exist it is necessary that
is the codimension of .
In the special case that (1) is the trivial real vector bundle , the notion of twisted normal framing (Def. ) reduces to that of normal framing.
The twisted Pontrjagin theorem identifies cobordism classes of twisted-framed submanifolds in (Def. ) with the twisted Cohomotopy of .
James Cruickshank, Twisted Cobordism and its Relationship to Equivariant Homotopy Theory, 1999 (pdf, pdf)
James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)
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