manifolds and cobordisms
cobordism theory, Introduction
(see also Chern-Weil theory, parameterized homotopy theory)
Consider a differentiable/smooth manifold $M^D$ of dimension $D$ and equipped with a real vector bundle
of rank $n \leq D$, with classifying map to be denoted $\tau \;\colon\; M \longrightarrow B \mathrm{O}(n)$.
Then a $\tau$-twisted normal framing on a submanifold $\Sigma^d \hookrightarrow M$ is an isomorphism of real vector bundles
between the normal bundle of the submanifold and the pullback bundle of $\mathcal{V}$ 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 $\Sigma^d \hookrightarrow M^D$.
In the special case that $\mathcal{V}$ (1) is the trivial real vector bundle $n$, 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 $M^D$ (Def. ) with the twisted Cohomotopy of $M^d$.
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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