nLab normal twisted framing

Definition

Definition

Consider a differentiable/smooth manifold $M^D$ of dimension $D$ and equipped with a real vector bundle

(1)$$\mathcal{V} \to M^d \;\;\;\; \in VectorBundles_{\mathbb{R}}(M)$$

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

$$N\Sigma \overset{\simeq}{\longrightarrow} \mathcal{V}_{\vert \Sigma}$$

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

$$n \;=\; D - d$$

is the codimension of $\Sigma^d \hookrightarrow M^D$.

Example

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.

Properties

Twisted Pntrjagin theorem

The twisted Pontrjagin theorem identifies cobordism classes of twisted-framed submanifolds in $M^D$ (Def. ) with the twisted Cohomotopy of $M^d$.

References

• 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)