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The regular preimage of the collapse recovers the original framed submanifold

Statement

Assume ${A C}_{ω}$ . Let $(N, φ)$ be a closed framed codimension- $k$ submanifold of a closed smooth $X$ , with $k \geq 1$ . For the normalized smooth Pontryagin–Thom representative $f$ of The Pontryagin-Thom map of a framed submanifold, its centre $y_{0}$ is regular, $f^{- 1} (y_{0}) = N$ , and the basis $b_{0}$ fixed there induces exactly $φ$ . Thus the framed regular preimage is $(N, φ)$ on the nose. Arbitrary unnormalized collapses represent the same homotopy class but need not induce this literal framing.

Facts & Assumptions

Given: $(N, φ)$ , a compatible tube $Φ$ inducing the identity on the normal quotient, and the normalized smooth representative $f$ .

[F1]

With $u = φ_{s} (v)$ , the target coordinate of $f (Φ (s, v))$ is $u / a (∣ u ∣^{2})$ for $∣ u ∣ < r$ , with $a = 1$ near zero and positive before the cutoff; elsewhere the value is $\infty$ (The Pontryagin-Thom map of a framed submanifold).

[F2]

A compatible chart fixes $N$ and induces the identity on its normal quotient (Pontryagin–Thom collapse with specified normal data). The preimage framing is the differential on that quotient followed by the coordinate isomorphism determined by the target basis (Framed regular preimages of a map to a sphere).

Proof

1.1F1givenalgebra

Since $φ_{s}$ is invertible and $a$ is positive on the finite-value region, $u / a (∣ u ∣^{2}) = 0$ precisely when $v = 0$ . The remaining points map to $\infty \neq y_{0}$ . Hence $f^{- 1} (y_{0}) = N$ .

2.1F1F2step 1.1algebra∎

Near $v = 0$ the target coordinate is exactly $φ_{s} (v)$ . Its vertical derivative is $φ_{s}$ , and its derivative on $T_{s} N$ is zero. Compatibility of the tube identifies the vertical quotient with $ν (N)_{s}$ by the identity, so the normal derivative of $f$ in the $b_{0}$ coordinates is exactly $φ_{s}$ . It is surjective, proving regularity and $f_{*} b_{0} = φ$ .

Depends on

Used by

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Milnor, Topology from the Differentiable Viewpoint (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002) (standard reference, not scraped)