Whitehead torsion of a finite CW homotopy equivalence

Let $f:X\to Y$ be a homotopy equivalence of finite CW complexes. Choose a vertex $x\in X$ and, after taking a cellular representative still denoted $f$, put $y=f(x)$, a vertex of $Y$; set $\pi ={\pi }_1(Y,y)$. Then the based induced map $f_{\ast }:{\pi }_1(X,x)\to \pi$ is an isomorphism (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type). Other basepoints are compared by supplied paths in the independence theorem. Choose

• a cellular representative of $f$, again written $f:X\to Y$, and homotopies ensuring that the cellular representative is a homotopy equivalence (Cellular approximation for maps of CW pairs);
• universal covers $p:\tilde{X}\to X$ and $q:\tilde{Y}\to Y$, chosen points $\tilde{x},\tilde{y}$ above $x,y$, and the compatible lift $\tilde{f}:\tilde{X}\to \tilde{Y}$ with $\tilde{f}(\tilde{x})=\tilde{y}$ of the cellular representative (Based cellular chains of a universal cover as finite free right group-ring modules);
• the based cellular chains of Based cellular chains of a universal cover as finite free right group-ring modules, with one oriented lift chosen for every cell of $X$ and of $Y$.

Coefficient transport. Transport the right $\mathbb{Z}[{\pi }_1(X,x)]$-module structure on $C_{\ast }(\tilde{X})$ along the ring isomorphism $\mathbb{Z}[f_{\ast }]:\mathbb{Z}[{\pi }_1(X,x)]\to \mathbb{Z}[\pi ]$, so that $c\cdot \lambda :=c\cdot f_{\ast }^{-1}(\lambda )$ for $\lambda \in \mathbb{Z}[\pi ]$. By Universal-cover boundaries, maps and homotopies respect the right group-ring action the transported boundary and the chain map $C_{\ast }(\tilde{f})$ are right $\mathbb{Z}[\pi ]$-linear, and by A lifted finite CW equivalence has a contractible group-ring mapping cone $C_{\ast }(\tilde{f})$ is a chain homotopy equivalence of bounded complexes of finite based free right $\mathbb{Z}[\pi ]$-modules.

The class. Form the algebraic mapping cone with the differential recorded in The mapping cone of a chain map, $\mathrm{Cone}(C_{\ast }(\tilde{f}){)}_n=C_n(\tilde{Y})\oplus C_{n-1}(\tilde{X}),d(y,x)=(d^{\tilde{Y}}y+C_{n-1}(\tilde{f})x,-d^{\tilde{X}}x),$ carrying in each degree the displayed basis of $C_n(\tilde{Y})$ followed by that of $C_{n-1}(\tilde{X})$, so that the target summands come first. This complex is bounded, finite based free and contractible, and $\mathbb{Z}[\pi ]$ has invariant basis number, so the contraction torsion of Finite based free complexes and contraction torsion is defined for it, is independent of the chosen contraction by Contraction torsion does not depend on the contraction, and lands in ${\tilde{K}}_1(\mathbb{Z}[\pi ])$. Define $\tau (f):=\text{image of }\tau (\mathrm{Cone}(C_{\ast }(\tilde{f})))\text{ in }\mathrm{W}\mathrm{h}(\pi )=K_1(\mathbb{Z}[\pi ])\mathrm{/}⟨[\pm \gamma ]:\gamma \in \pi ⟩,$ using the quotient map of K₁ of a ring and the Whitehead group of a discrete group. Equivalently, $\tau (f)$ is the image of the reduced class $[(d+s{)}_{\mathrm{o}\mathrm{d}\mathrm{d}}]\in {\tilde{K}}_1(\mathbb{Z}[\pi ])$ for any contraction $s$ of the cone.

Disconnected complexes. If $Y$ has components $D$, write ${\pi }_{1}D$ for the fundamental group of a component at a chosen basepoint; every component of a finite CW complex has the homotopy type of a connected finite CW complex and contains a vertex, so this is defined. Since $f$ is a homotopy equivalence it maps components of $X$ bijectively onto components of $Y$, and the data above are chosen componentwise; the class $\tau (f)\in {⨁}_{D\in {\pi }_0(Y)}\mathrm{W}\mathrm{h}({\pi }_{1}D)$ has as its $D$-component the torsion of the restriction to the component of $Y$ corresponding to $D$, computed with a basepoint in that component and its chosen lift. For connected $Y$ this is the single class defined above.

Status. This is a definition by chosen data: the contraction, the cellular representative, the basepoint paths, the lifts, the orientations and the order of the cells are all auxiliary. The next theorem proves that the resulting class in $\mathrm{W}\mathrm{h}$ does not depend on them; until then $\tau (f)$ denotes the class attached to the displayed choices. No further quotient and no further choice principle is used, the cellular representative exists choice-free for finite complexes, and all choices made here are finite except the (finite) choice of cell lifts.