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Handle slides, renumberings and reorientations reduce a unimodular middle-handle matrix to the identity

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). (i) Every invertible integer matrix $A \in {GL}_{r} (Z)$ can be carried to the identity by finitely many operations of the following three kinds: add an integer multiple of one row (respectively column) to another row (respectively column); interchange two rows (respectively columns); multiply a row (respectively column) by $- 1$ . (ii) Consequently, if an h-cobordism as in the previous lemma is presented with handles only in indices $k, k + 1$ and middle-handle intersection matrix $M \in {GL}_{r} (Z)$ (Acyclicity makes the simply connected middle-handle matrix unimodular), then $W$ admits a presentation relative to $M_{0}$ with the same indices and the same number of handles whose middle-handle intersection matrix is the identity $I_{r}$ : each operation of (i) is realised by a handle slide, a renumbering of equal-index handles or a reorientation of a handle core or cocore, all of which preserve $W$ relative to $M_{0}$ .

Facts & Assumptions

Given: An integer matrix $A \in {GL}_{r} (Z)$ ; and an h-cobordism presented with handles only in indices $k, k + 1$ , $2 \leq k \leq n - 2$ , whose middle-handle intersection matrix is $M \in {GL}_{r} (Z)$ ; ${A C}_{ω}$ .

[F1]

The Smith normal form existence theorem over the PID $Z$ supplies a matrix-equivalent diagonal matrix $diag (d_{1}, \dots, d_{r})$ with $d_{1} ∣ \dots ∣ d_{r}$ and the elementary implementation needed here is proved directly in step 1.1 below; over $Z$ the units are $\pm 1$ (Every matrix over a PID has a Smith normal form, Matrix equivalence and Smith normal form over a PID).

[F2]

The determinant of a diagonal matrix is the product of its diagonal entries, so if $A$ is invertible with $\det A = \pm 1$ then $\prod_{i} d_{i} = \pm 1$ and each $d_{i} = \pm 1$ ; sign changes turn the diagonal matrix into $I_{r}$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[F3]

Under the disk-push comparison together with its specified lower-stage homotopy, a slide changes an upper or lower core basis generator by adding $\pm$ another generator, and preserves the relative diffeomorphism type. Handle slides act by elementary basis change on handle chains, Handle slides preserve the relative diffeomorphism type

[F4]

Rows index upper handles and columns index lower handles; the differential acts on row vectors by $x \mapsto x M$ . Relabeling or reorienting a core relabels or changes the sign of its basis generator. The middle-handle intersection matrix of an h-cobordism

Proof

technique · direct

1.1F1F2givenalgebra

A finite elementary reduction exists directly over $Z$ . For $r = 0$ there is nothing to do. For $r \geq 1$ , the entries of the first column of $A$ have gcd $1$ : the first row of the integer inverse of $A$ supplies an integer linear combination equal to $1$ . Apply the Euclidean algorithm to pairs of entries, using row swaps and subtraction of integer multiples; every nonzero remainder is smaller in absolute value than the previous divisor, so each pair reduction terminates. Iterating through the finite column yields $(1, 0, \dots, 0)^{T}$ , with a final sign change if needed. Then subtract suitable multiples of column one from the other columns to clear the first row. The resulting matrix is $diag (1, B)$ with $B \in {GL}_{r - 1} (Z)$ , because the performed operations are invertible over $Z$ . Repeat on $B$ ; induction on the matrix size terminates with $I_{r}$ . All operations are exactly additions, swaps and sign changes, proving (i). This also gives the required elementary implementation of the Smith-form conclusion of [F1].

2.1F3F4step 1.1algebra

For an upper slide $g_{i^{'}}^{'} = g_{i^{'}} + ε g_{i}$ , linearity gives $R_{i^{'}}^{'} = R_{i^{'}} + ε R_{i}$ . For a lower slide $e_{j^{'}}^{'} = e_{j^{'}} + ε e_{j}$ , rewrite each boundary in the new basis: $e_{j^{'}} = e_{j^{'}}^{'} - ε e_{j}^{'}$ . Thus $C_{j}^{'} = C_{j} - ε C_{j^{'}}$ and the other columns are unchanged. Choosing the opposite slide direction realizes any desired target column addition; finitely repeating a unit addition realizes any integer multiple. In ambient dimension $n + 1$ , both handle indices $k, k + 1$ lie between $1$ and $(n + 1) - 2 = n - 1$ , so [F3] applies throughout $2 \leq k \leq n - 2$ . Swaps and sign changes are relabelings and reorientations by [F4].

3.1F4givenstep 2.1∎

Apply to the presentation of (ii) the finite sequence of handle slides, renumberings and reorientations corresponding to the algebraic sequence of step 1.1: each step preserves $W$ relative to $M_{0}$ , so the final presentation has the same indices and the same number of handles and its middle-handle intersection matrix is $I_{r}$ by [F1] and the definition of the matrix as the differential $\partial_{k + 1}$ in the handle bases.

Depends on

Used by

Dependency tree · two levels

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Sources

Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, Oxford University Press 2002; complete electronic copy) (standard reference, not scraped)
Wolfgang Lück, A Basic Introduction to Surgery Theory (ICTP lecture notes, 27 October 2004; complete author text) (standard reference, not scraped)