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A free-face interval expansion has zero torsion

Statement

Starting from the one-vertex complex $X = {v}$ , attach a new vertex $w$ and a $1$ -cell $e$ running from $v$ to $w$ . The new vertex is a free face of the new $1$ -cell, so $X ↪ Y$ is an elementary expansion. The relative universal-cover chains are $0 \to Z ⟨ e ⟩ \overset{1}{\to} Z ⟨ w ⟩ \to 0,$ so that $τ (X ↪ Y) = 0$ in $W h (1)$ . The same computation over any group $π$ produces the boundary unit $\pm g$ and the zero Whitehead class.

Facts & Assumptions

Given: The one-vertex complex $X = {v}$ , a new vertex $w$ and a new $1$ -cell $e$ attached from $v$ to $w$ , giving $Y = X \cup {w} \cup e$ .

[F1]

An elementary expansion of dimension $n$ is an inclusion $X ↪ Y$ together with a homeomorphism $Φ : (D^{n}, D_{+}^{n - 1}) \to (Q^{n}, Q^{n - 1})$ and a characteristic map $φ : Q^{n} \to Y$ such that $φ ∣_{Q^{n - 1}}$ is a characteristic map for the new $(n - 1)$ -cell $e^{n - 1}$ , all other boundary values of $φ$ lie in $X$ , and $Y = X \cup e^{n - 1} \cup e^{n}$ ; equivalently the new $(n - 1)$ -cell is a free face of the new $n$ -cell, and the pair deformation retracts onto $X$ (Elementary expansions and collapses of finite CW complexes, Cell attachment by a characteristic map).

[F2]

The relative cellular chains of a finite CW pair over the universal cover with chosen oriented lifts are finite free right $Z [π_{1}]$ -modules on the lifts of the relative cells, concentrated in the degrees in which relative cells occur, and the boundary is right group-ring linear (Based cellular chains of a universal cover as finite free right group-ring modules, Universal-cover boundaries, maps and homotopies respect the right group-ring action).

[F3]

If $j : X ↪ Y$ is an elementary expansion of finite CW complexes, then $τ (j) = 0$ in $W h (π_{1} Y)$ , componentwise, and the only nonzero relative cellular boundary of the pair in suitable oriented lifts is $R \overset{\pm g}{\to} R$ in two consecutive degrees, with $g \in π_{1} Y$ and $[\pm g] = 0$ in $W h (π_{1} Y)$ (An elementary CW expansion has zero Whitehead torsion).

[F4]

$τ$ of a homotopy equivalence of finite CW complexes is the image in the target Whitehead group of the contraction torsion of its algebraic mapping cone; $K_{1}$ is written additively with $[I] = 0$ and $[A^{- 1}] = - [A]$ , and ${\tilde{K}}_{1} (R) = K_{1} (R) / ⟨ [- 1] ⟩$ , so both $[1]$ and $[- 1]$ vanish in ${\tilde{K}}_{1} (R)$ (Whitehead torsion of a finite CW homotopy equivalence, K₁ of a ring and the Whitehead group of a discrete group).

[F5]

Every nonempty convex subset of $R^{n}$ is contractible, a contractible space has trivial fundamental group at each basepoint, and the induced maps on all homotopy groups are functorial and invariant under based homotopies (Every nonempty convex subset of $R^{n}$ is contractible, A contractible space has trivial fundamental group, Higher homotopy groups are functorial and based homotopy invariant).

Proof

technique · direct

1.1

Realize $Y$ as the CW complex with vertex set ${v, w}$ and the single $1$ -cell $e$ attached by the map $q : S^{0} = {- 1, + 1} \to Y^{0}$ with $q (- 1) = v$ and $q (+ 1) = w$ . With $Q^{1} = D^{1} = [- 1, 1]$ , $Q^{0} = D_{+}^{0} = {+ 1}$ and $φ$ the characteristic map of $e$ , the restriction $φ ∣_{Q^{0}}$ is a characteristic map for the new $0$ -cell $w$ (mapping $+ 1$ homeomorphically onto $w$ ), the remaining boundary value $φ (- 1) = v$ lies in $X$ , and $Y = X \cup {w} \cup e$ . Hence $X ↪ Y$ is an elementary expansion of dimension $1$ by [F1], and the new vertex $w$ is a free face of $e$ .

1.2

The pair $(Y, X)$ deformation retracts onto $X$ by [F1]; the retraction $r : Y \to X$ therefore satisfies $r \circ i = {i d}_{X}$ for the inclusion $i : X = {v} ↪ Y$ and $i \circ r ≃ {i d}_{Y}$ . The one-point space $X$ is a nonempty convex subset of $R^{1}$ , hence contractible with $π_{1} (X, v) = 1$ by [F5], and functoriality together with based-homotopy invariance of the induced maps gives $i_{*} \circ r_{*} = (i \circ r)_{*} = i d$ and $r_{*} \circ i_{*} = (r \circ i)_{*} = i d$ on $π_{1}$ , so $i_{*} : π_{1} (X, v) \to π_{1} (Y, v)$ is an isomorphism and $π_{1} (Y, v) = 1$ ; the edge $e$ supplies a path from $v$ to $w$ , so basepoint change also gives $π_{1} (Y, w) = 1$ .

F1F5

1.3

Over the universal cover the pair $(Y, X)$ has exactly two relative cells, the lift of $w$ in degree $0$ and the lift of $e$ in degree $1$ , so by [F2] the relative based chain complex $T_{∙}$ is concentrated in degrees $0$ and $1$ with a single displayed basis vector in each degree and its differential has the $1 \times 1$ right-module matrix $\pm 1$ : the characteristic map restricts to a homeomorphism on the free face, so the incidence of $w$ in the boundary of the lift of $e$ is $\pm 1$ . This is the complex displayed in the statement, where the orientation is chosen so that the entry is $1$ .

F1F2

2.1

With the contraction $s (e_{0}) = e_{1}$ for the displayed differential $d (e_{1}) = e_{0}$ , the odd-to-even map $(d + s)_{o d d} : T_{1} \to T_{0}$ is $d$ and has the $1 \times 1$ matrix $(1)$ . Thus $τ (T_{∙}) = [1] = 0$ in ${\tilde{K}}_{1} (Z)$ . Reversing either cell orientation changes the matrix to $(- 1)$ , whose class is also $0$ in the reduced group. The resulting image in $W h (1)$ vanishes.

F2F4step 1.3

2.2

By [F3] applied to the elementary expansion $j : X ↪ Y$ of step 1.1, $τ (X ↪ Y) = 0$ in $W h (π_{1} Y) = W h (1)$ by step 1.2; the relative boundary is the unit $\pm g$ of [F3] with $g$ the group element determined by the chosen lifts, and $[\pm g] = 0$ in $W h (π_{1} Y)$ since here $g = 1$ .

F3step 1.1step 1.2

3.1

The identical computation applies to an elementary expansion performed on any finite CW complex: by [F3] the only nonzero relative cellular boundary of the pair is the unit $\pm g$ in two consecutive degrees and the class $[\pm g]$ dies in $W h (π_{1} Y)$ , so the boundary unit $\pm g$ produces zero Whitehead class over any group $π$ .

F3∎

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Sources

Lück, Lemma 2.18(1), p.35 (standard reference, not scraped)
Davis–Kirk, Theorem 11.31(2), p.344 (standard reference, not scraped)