TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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Whitehead torsion is independent of all auxiliary choices

Statement

Let $f : X \to Y$ be a homotopy equivalence of finite CW complexes and let $τ (f)$ be the class of Whitehead torsion of a finite CW homotopy equivalence attached to a choice of cellular representative, universal covers, lifts, basepoints, orientations, orders of the cells and chain contraction. Then the image of that class in $W h (π_{1} (Y, y))$ depends on none of these choices.

Moreover:

Changing the basepoint $y$ to $y^{'}$ transports $τ (f)$ under the canonical isomorphism $W h (π_{1} (Y, y)) \to W h (π_{1} (Y, y^{'}))$ of basepoint change, and two different paths from $y$ to $y^{'}$ induce the same isomorphism, so the transport is canonical; the same holds at the source.
If $f^{'} ≃ f$ is a second finite CW homotopy equivalence with the same target basepoint, then $τ (f^{'}) = τ (f)$ after the canonical identifications; in particular homotopic homotopy equivalences of finite CW complexes have equal torsion.
For disconnected $Y$ all statements hold componentwise in $⨁_{D \in π_{0} (Y)} W h (π_{1} D)$ .

Facts & Assumptions

Given: A homotopy equivalence $f : X \to Y$ of finite CW complexes with the chosen data of Whitehead torsion of a finite CW homotopy equivalence, and a second choice of the same kind, written with primes.

[F1]

$τ (f)$ is the image in $W h (π_{1} (Y, y))$ of the contraction torsion of the based contractible complex $Cone (C_{*} (\tilde{f}))_{n} = C_{n} (\tilde{Y}) \oplus C_{n - 1} (\tilde{X})$ with the target summands first, computed from the odd-to-even part of $d + s$ in the displayed bases (Whitehead torsion of a finite CW homotopy equivalence, Finite based free complexes and contraction torsion).

[F2]

The contraction torsion of a bounded contractible based free complex does not depend on the contraction (Contraction torsion does not depend on the contraction).

[F3]

If the degree- $n$ basis is replaced by the basis whose coordinate columns are the columns of the invertible matrix $P_{n}$ in the old basis, then $τ_{new} = τ_{old} + \sum_{n} (- 1)^{n + 1} [P_{n}]$ in ${\tilde{K}}_{1} (R)$ ; if $u : F \to G$ is a chain isomorphism with equal degreewise displayed basis sizes then $τ (G) = τ (F) + \sum_{n} (- 1)^{n} [u_{n}]$ ; and unipotent upper triangular matrices have class $0$ (Basis-change, direct-sum and based exact-sequence formulas, Stable elementary matrices equal the commutator subgroup).

[F4]

Reordering a basis changes the class in $K_{1}$ only by $[- 1]$ , reversing an orientation or replacing a chosen cell lift changes it by a unit $\pm g$ of $Z [π]$ , and all of these classes vanish in $W h (π)$ ; a change of basepoint path conjugates the fundamental group, inner automorphisms induce the identity on $W h$ , and two basepoint paths induce the same map there (Cellular basis ambiguities vanish in the Whitehead group).

[F5]

Chain homotopic chain maps $f_{0} ≃ f_{1} : C_{*} \to D_{*}$ have chain-isomorphic mapping cones, by the unitriangular isomorphism $Ψ (y, x) = (y + h_{n - 1} x, x)$ on $D_{n} \oplus C_{n - 1}$ for a chain homotopy $h$ ; chain homotopy is compatible with composition, and a deck transformation is an additive chain isomorphism which is semilinear for the corresponding inner automorphism of the group ring (Homotopic maps have chain-isomorphic mapping cones, The mapping cone of a chain map, Chain homotopy is compatible with addition and composition, Universal-cover boundaries, maps and homotopies respect the right group-ring action).

[F6]

A continuous map of finite CW complexes is homotopic to a cellular map, and homotopic cellular maps of finite CW pairs admit a cellular homotopy; both statements are choice-free for finite complexes (Cellular approximation for maps of CW pairs).

[F7]

A lifted cellular homotopy induces a right-linear chain homotopy from the initial compatible lift to the deck-twisted endpoint composite, both interpreted with the initial map’s coefficient transport; the deck map alone need not be right-linear (Universal-cover boundaries, maps and homotopies respect the right group-ring action).

Proof

technique · direct

1.1

Replacing the contraction of the cone changes nothing, by [F2]; this proves independence of the contraction.

F1F2

1.2

Consider a change of the displayed cellular bases only. In degree $n$ the cone basis is the concatenation of the basis of $C_{n} (\tilde{Y})$ and of $C_{n - 1} (\tilde{X})$ , so a change of the cell bases induces the block-diagonal basis change $diag (P_{n}^{\tilde{Y}}, P_{n - 1}^{\tilde{X}})$ in degree $n$ ; by [F3] the torsion changes by $\sum_{n} (- 1)^{n + 1} [diag (P_{n}^{\tilde{Y}}, P_{n - 1}^{\tilde{X}})]$ . Reordering, reorienting or relifting cells changes only the individual factors $P_{n}^{\tilde{Y}}$ , $P_{n - 1}^{\tilde{X}}$ by permutation matrices, by diagonal matrices with a single entry $- 1$ , or by diagonal matrices with a single entry a group element, all of which have class $0$ in $W h (π_{1} (Y, y))$ by [F4]; hence the class in $W h$ is unchanged.

F1F3F4

1.3

For a fixed based map $f$ and based cover identifications, a compatible lift is unique. Changing the chosen lift over the target basepoint by a deck map $T_{β}$ changes the compatible lift to $T_{β} \tilde{f}$ and the coefficient identification from $f_{*}$ to $α_{β} f_{*}$ , where $α_{β} (h) = β h β^{- 1}$ . Indeed $T_{β} (c \cdot h) = T_{β} (c) \cdot α_{β} (h)$ , so $T_{β}$ is $α_{β}$ -semilinear, generally not $R$ -linear. With this simultaneous coefficient change, $(y, u) \mapsto (T_{β} y, u)$ is a semilinear chain isomorphism from the cone of $C_{*} (\tilde{f})$ to the cone of $C_{*} (T_{β} \tilde{f})$ . In the transported target bases $T_{β} \tilde{e}$ and unchanged source bases it preserves the displayed bases; using the original target lifts instead changes each target basis by a diagonal group unit. Inner automorphisms act trivially on $W h$ and these diagonal units vanish there by [F4]. Thus the torsion class is unchanged under a change of compatible cover identification or lifted map.

F1F3F4F5

1.4

A change of basepoint $y \to y^{'}$ transports the coefficient group ring along the basepoint isomorphism $π_{1} (Y, y) \to π_{1} (Y, y^{'})$ of [F4], and every path gives the same map on $W h$ because two such isomorphisms differ by an inner automorphism, which acts trivially; the same argument applies at the source, and the class of a componentwise definition on a disconnected target is transported componentwise.

2.1

Let $f^{'}$ be a second cellular representative of the given homotopy class, with compatible lift ${\tilde{f}}^{'}$ . By [F6] there is a cellular homotopy $H$ from $f$ to $f^{'}$ . Its lift beginning at $\tilde{f}$ ends at $T_{β} {\tilde{f}}^{'}$ , and [F7] supplies a right-linear chain homotopy $C_{*} (\tilde{f}) ≃ C_{*} (T_{β} {\tilde{f}}^{'})$ for the initial coefficient transport. Step 1.3 identifies the torsion of the latter map, after its corresponding inner coefficient transport, with the torsion of $C_{*} ({\tilde{f}}^{'})$ . Thus it remains to compare cones of the two chain-homotopic maps over the same ring.

F6F7step 1.3

3.1

For chain-homotopic $f_{0} ≃ f_{1}$ the isomorphism $Ψ$ of [F5] has matrix $(\begin{array}{cc} I & h_{n - 1} \\ 0 & I \end{array})$ in the target-first cone bases, so by the isomorphism formula of [F3] the two cone torsions differ by a sum of classes of unipotent matrices, namely $0$ ; hence the two cones give the same class in ${\tilde{K}}_{1}$ and therefore the same class in $W h$ . This proves independence of the cellular representative and of homotopic replacements.

F3F5step 2.1

4.1

Combining steps 1.1, 1.2, 1.3 and 1.4 gives independence of contraction, cell bases, cover identifications, lifts and basepoint paths; combining with step 3.1 gives independence of the cellular approximation and equality for homotopic homotopy equivalences; the disconnected statement is the componentwise reading of steps 1.1 through 1.4 and step 2.1.

step 1.1step 1.2step 1.3step 1.4step 3.1∎

Depends on

Used by

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Sources

Lück, Definition 2.13 and Lemma 2.9, pp.29–31 (standard reference, not scraped)
Davis–Kirk, Theorem 11.31(1), pp.343–344 (standard reference, not scraped)
Cohen, §§19,22, pp.62–65,72–75 (standard reference, not scraped)