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Vanishing torsion allows algebraic diagonalization by simple handle moves

Statement

Assume ${A C}_{ω}$ . Let a nonempty connected oriented smooth h-cobordism of dimension $n + 1 \geq 6$ have a two-index presentation in degrees $q, q + 1$ , $2 \leq q \leq n - 2$ , with right-module differential matrix $A \in {G L}_{c} (R)$ , $R = Z [π_{1} (W)]$ . If $[A] = 0$ in $Wh (π_{1} (W))$ , finitely many cancelling-pair additions, equal-index slides, reorderings and choices of oriented lifts give a presentation with diagonal differential matrix of size $c + b$ and entries $\pm g_{i}$ , for some $b \geq 0$ . The stabilization is allowed to remain until geometric cancellation.

Algebraically there are elementary matrices $E_{1}, \dots, E_{k}$ of size $c + b$ and a trivial-unit diagonal matrix $D$ with $A \oplus I_{b} = E_{1} \dots E_{k} D .$

Facts & Assumptions

Given: The oriented high-dimensional two-index presentation and $[A] = 0$ in the statement.

[F1]

$K_{1} (R) = G L (R) / E (R)$ and $Wh (π) = K_{1} (R) / ⟨ [\pm g] ⟩$ ; the stable elementary subgroup is normal. K₁ of a ring and the Whitehead group of a discrete group, Stable general linear and elementary groups for right modules, Stable elementary matrices equal the commutator subgroup.

[F2]

A cancelling $q / (q + 1)$ pair adds a unit block, normalized to $1$ by orientations and lifts. Elementary and trivial-unit basis changes have zero Whitehead class. Creation of a cancelling handle pair, Cellular basis ambiguities vanish in the Whitehead group, Handle slides and cancelling-pair creations preserve Whitehead torsion.

[F3]

The modification construction in the complement of one omitted upper handle replaces its attaching-sphere class $v_{j}$ by $v_{j} + v_{i} r$ , $i \neq j$ , and gives an isotopy after the other upper handles are attached. It preserves its framing and the resulting manifold. The group-ring modification lemma for embedded spheres, Isotopic attaching embeddings give diffeomorphic handle attachments.

[F4]

A two-term differential in degrees $q + 1, q$ has contraction torsion $(- 1)^{q} [A]$ ; its right-module matrix uses lower handles as rows and upper handles as columns. The based handle chain complex over the fundamental group ring, The torsion of the handle complex is the torsion of the inclusion.

Proof

1.1F1given

By [F1], the $K_{1}$ class of $A$ is a finite sum of classes of units $\pm g$ , with inverses again of that form. Represent that sum by a finite diagonal matrix $D$ , padding $A$ and $D$ by identities to the same size. Then $(A \oplus I) D^{- 1} \in E (R)$ . Membership in the stable union gives a further finite padding for which it is a finite product of elementary matrices. This proves the displayed factorization; $D$ has $c + b$ diagonal entries.

2.1F1F2step 1.1

Normalize the target handle basis by $D$ . In right coordinate columns, a new target basis with matrix $D$ changes the differential $B = A \oplus I_{b}$ to $D^{- 1} B$ . Normality in [F1] gives $D^{- 1} B \in E (R)$ ; after further identity padding if necessary it is a product of elementary matrices of the displayed size. The padding is geometrically supplied by [F2], and each diagonal basis change is an orientation or deck-lift choice.

3.1F3F4step 2.1construct

Realize multiplication on the right by an elementary matrix $e_{i j} (r)$ . Its effect is column $j \mapsto$ column $j +$ column $i \cdot r$ . Apply [F3] with the $j$ th upper handle omitted and the $i$ th retained. All upper handles have index $q + 1 \geq 3$ , so omitting one changes no fundamental group; thus the ambient coefficient group is still $π_{1} (W)$ . The framed band modifications of [F3] are upper-handle slides, one per signed monomial of $r$ ; arbitrary $r$ generally needs several slides. The new attaching embedding is isotopic after the retained upper handles are attached, so the relative diffeomorphism type is preserved. No incorrect direct identification of core and belt basis changes is used.

4.1F2F4step 2.1step 3.1∎

Apply step 3.1 to the elementary factors of $(D^{- 1} B)^{- 1}$ in their multiplication order. The resulting matrix is $I_{c + b}$ , a permitted trivial-unit diagonal matrix. All its added handles remain part of the stabilized presentation. By [F2] torsion is unchanged, and [F4] gives $(- 1)^{q} [I_{c + b}] = 0$ . This proves the geometric diagonalization with the stated dimension and stabilization hypotheses.

Depends on

Used by

Vanishing presentation-indexed torsion implies the product cobordism Theorem

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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)