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Vanishing presentation-indexed torsion implies the product cobordism

Statement

Assume ${A C}_{ω}$ . Let $(W; M_{0}, M_{1})$ be a nonempty connected oriented smooth h-cobordism of dimension $n + 1 \geq 6$ with $M_{0}$ closed connected oriented and let $π = π_{1} (M_{0})$ . If a finite handle presentation $H$ of $(W, M_{0})$ has $τ_{H} (W, M_{0}) = 0$ , then $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ . This applies to any finite presentation, not only one already in two-index normal form. The orientation hypothesis is exactly the one under which the locally proved oriented intersection-matrix route applies; no orientation-free strengthening is claimed.

Facts & Assumptions

Given: A nonempty connected oriented smooth h-cobordism $(W; M_{0}, M_{1})$ of dimension $n + 1 \geq 6$ with $M_{0}$ closed connected oriented, and a finite handle presentation $H$ of $(W, M_{0})$ with $τ_{H} (W, M_{0}) = 0$ .

[F1]

Every finite handle presentation can be put into two-index normal form at any index $2 \leq q \leq n - 2$ for the oriented data of the statement (the hypotheses of the normal-form lemma), with handles only in degrees $q, q + 1$ and invertible intersection matrix, by the elementary handle modifications of the previous lemma, which preserve the presentation-indexed torsion (h-cobordisms admit two-index normal form presentations, Handle slides and cancelling-pair creations preserve Whitehead torsion, Presentation-indexed Whitehead torsion of an h-cobordism).

[F2]

In a two-index presentation at index $q$ the based relative complex is the two-term complex with differential the intersection matrix $A$ , and the presentation-indexed torsion is $(- 1)^{q} [A]$ ; hence a vanishing torsion class gives $[A] = 0$ in $Wh (π)$ (The based handle chain complex over the fundamental group ring, Presentation-indexed Whitehead torsion of an h-cobordism).

[F3]

A matrix with vanishing Whitehead class can be diagonalized by elementary basis changes, cancelling-pair stabilizations and unit changes, each realized geometrically by simple handle moves, and a diagonal presentation can be isotoped into cancelling position and cancelled to the empty presentation, which exhibits the product (Vanishing torsion allows algebraic diagonalization by simple handle moves, Group-labelled Whitney tricks realize the diagonalized handle complex, A cobordism with no handles is a product, h-Cobordism).

Proof

1.1F1given

Put the given presentation $H$ into two-index normal form at the index $q = 2$ , which is allowed because $2 \leq q \leq n - 2$ holds for $n \geq 5$ : by [F1] the resulting presentation $H^{'}$ has handles only in degrees $2$ and $3$ , with invertible intersection matrix $A$ , and the elementary modifications preserve the torsion class, so $τ_{H^{'}} (W, M_{0}) = τ_{H} (W, M_{0}) = 0$ .

2.1F2step 1.1

By [F2] applied with $q = 2$ the class of the intersection matrix satisfies $[A] = (- 1)^{2} τ_{H^{'}} (W, M_{0}) = 0$ in $Wh (π)$ .

3.1F3step 2.1

By [F3] the vanishing class of $A$ allows the matrix to be diagonalized with unit diagonal entries by elementary operations and cancelling-pair stabilizations, all realized by simple handle moves, and the resulting diagonal presentation can be isotoped so that each $(q + 1)$ -handle attaches in cancelling position to its $q$ -handle, after which the pairs are cancelled; the final presentation has no handles.

4.1F3step 3.1∎

A presentation of $(W, M_{0})$ with no handles shows by [F3] that $W$ is diffeomorphic to $M_{0} \times [0, 1]$ relative to $M_{0}$ . The argument applies to the given arbitrary finite presentation, since the passage to normal form used only the allowed modifications.

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Sources

Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, electronic edition) (standard reference, not scraped)