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Zero relative torsion gives a finite relative elementary deformation

Statement

Let $L \subset K$ be a homotopy-equivalence inclusion of connected finite CW complexes. If $τ (L ↪ K) = 0$ in $Wh (π_{1} K)$ , then $K$ is carried to $L$ by finitely many elementary expansions and collapses fixing $L$ , with only reorderings, orientation reversals and deck-lift changes of the cellular bases. This is the geometric converse for inclusions.

Facts & Assumptions

Given: The finite homotopy-equivalence inclusion with zero Whitehead torsion.

[F1]

Finite relative cell trading fixes $L$ , transports torsion and leaves cells only in two degrees $n, n + 1$ with $n \geq 3$ (Cell trading puts a finite relative equivalence in two high degrees).

[F2]

In that two-layer pair the relative cellular differential is an invertible matrix $A$ in the group-ring homotopy bases (Two high relative cell layers have free homotopy bases and their cellular boundary matrix).

[F3]

Finite relative elementary moves realize left and right stable elementary matrix operations, identity-block stabilization, and the listed trivial basis changes (Cell slides and stabilizations realize elementary group-ring matrices).

[F4]

An identity relative homotopy matrix permits cancellation of all relative cells by finite elementary moves (An identity relative homotopy matrix permits cell cancellation).

[F5]

For $R = Z [π]$ , $Wh (π) = G L (R) / (E (R) ⟨ \pm g : g \in π ⟩)$ , in additive $K_{1}$ notation (K₁ of a ring and the Whitehead group of a discrete group).

[F6]

The torsion of a homotopy-equivalence inclusion is the based relative universal-cover chain torsion, and for a two-term complex in degrees $n, n + 1$ its class is $(- 1)^{n + 2} [A]$ in $Wh (π)$ (Whitehead torsion of a finite CW homotopy equivalence).

[F7]

A simple deformation has zero torsion and the composition formula transports inclusion torsion along it (Simple homotopy equivalences have zero torsion, Composition and based-pair sum formulas for Whitehead torsion).

Proof

technique · direct

1.1

Apply [F1] to obtain a two-high-layer pair $(K^{'}, L)$ and a simple homotopy equivalence $h : K \to K^{'}$ fixing $L$ . By [F7], $τ (L ↪ K^{'}) = h_{*} τ (L ↪ K) = 0$ ; the group isomorphism induced by $h$ transports this equality without selecting a new generator.

F1F7given

2.1

Choose the finite lower and upper lifted-cell bases of [F2]. The cellular differential is $A \in G L_{a} (R)$ , including the empty $0 \times 0$ matrix when $a = 0$ . By [F6] its torsion is $(- 1)^{n + 2} [A]$ , so zero torsion implies $[A] = 0$ in $Wh (π)$ . The parity sign has no effect on vanishing.

F2F6step 1.1

3.1

By the definition of the quotient [F5], there is a finite diagonal block $T = diag (ϵ_{1} g_{1}, \dots, ϵ_{b} g_{b})$ with $ϵ_{j} \in {\pm 1}$ and $g_{j} \in π$ , and a stabilization size $m$ , such that $diag (A, I_{m}) diag (T^{- 1}, I)$ lies in the stable elementary subgroup. Equivalently, after a common finite stabilization, $A$ differs from a product of elementary matrices by finitely many trivial units. This uses equality in the direct limit $G L (R)$ , so the stabilization is finite; it does not assert that an arbitrary unit of $R$ is trivial.

F5step 2.1

4.1

Apply [F3] for the identity-block stabilization and for the finite elementary factors in the inverse order. Absorb each diagonal $ϵ_{j} g_{j}$ by changing the orientation or chosen deck lift of its corresponding cell. The resulting relative boundary matrix is the identity in the transported characteristic bases. Every move is a finite expansion or collapse relative to $L$ , and each basis change is merely a change of description of the same cells.

F3step 3.1

5.1

Apply [F4] to the resulting identity-matrix pair. It cancels all relative cells by finite elementary moves fixing $L$ . Concatenating this deformation with those of steps 1.1 and 4.1 gives the required finite formal deformation of $K$ to $L$ . If $a = 0$ , [F4] is the empty deformation and the same conclusion holds. ∎

F4step 1.1step 4.1

Depends on

Used by

Whitehead torsion is the complete obstruction to finite CW simple homotopy Theorem

Dependency tree · two levels

51 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

Cohen, §§7.3–8.5, printed pp.25–33 (standard reference, not scraped)
Casson, Theorem 4.7, printed pp.32–34 (standard reference, not scraped)
Lück, Theorem 2.21, printed pp.37–38 (standard reference, not scraped)