[proofplan] We first replace $f$ by a cellular representative and compute its torsion from the cellular mapping cone over $\mathbb Z[G]$. A simple homotopy equivalence is built from elementary expansions, elementary collapses, cellular isomorphisms, and homotopies, and each such elementary move contributes only elementary matrices and elementary acyclic summands, hence has zero class in $\operatorname{Wh}(G)$. Conversely, if $\tau(f)=0$, then the mapping cone is not only acyclic but simply contractible as a finite based free $\mathbb Z[G]$-complex. The cellular realization theorem for simple chain contractions then realizes this algebraic simple contraction by elementary expansions and collapses, so $f$ is simple. [/proofplan] [step:Replace the map by a cellular representative and fix the chain-level torsion convention] Choose a cellular map $f_c:X\to Y$ homotopic to $f$. This uses the cellular approximation theorem, in the form that every map between CW-complexes is homotopic to a cellular map. By the homotopy-invariance and choice-independence convention included in the definition of $\tau(f)$ in the statement, $\tau(f)=\tau(f_c)$, and $f$ is a simple homotopy equivalence if and only if $f_c$ is a simple homotopy equivalence. Choose universal covers $p_Y:\widetilde Y\to Y$ and $p_X:\widetilde X\to X$, and use the isomorphism $f_*:\pi_1(X,x_0)\to G$ to regard the cellular chain complex $C_*(\widetilde X)$ as a based finite free left $\mathbb Z[G]$-chain complex. Choose a lift $\widetilde f_c:\widetilde X\to \widetilde Y$ of $f_c$. It induces a chain map \begin{align*} C_*(\widetilde f_c):C_*(\widetilde X)\to C_*(\widetilde Y) \end{align*} of based finite free left $\mathbb Z[G]$-chain complexes. Since $f_c$ is a homotopy equivalence and $f_*:\pi_1(X,x_0)\to G$ is an isomorphism, choose a homotopy inverse $g_c:Y\to X$ after replacing it by a cellular representative if necessary. The homotopies $g_c\circ f_c\simeq \operatorname{id}_X$ and $f_c\circ g_c\simeq \operatorname{id}_Y$ induce the identity on the relevant fundamental groups, so after choosing compatible basepoints in the universal covers they lift to homotopies between the corresponding lifted composites and the identity maps on $\widetilde X$ and $\widetilde Y$. Hence $\widetilde f_c$ is a homotopy equivalence of universal covers. Therefore the cellular mapping cone \begin{align*} \operatorname{Cone}(C_*(\widetilde f_c)) \end{align*} is an acyclic based finite free $\mathbb Z[G]$-chain complex. By definition, $\tau(f_c)$ is the torsion of this mapping cone, viewed in $\operatorname{Wh}(G)$. [/step] [step:Show that elementary expansions and collapses have zero Whitehead torsion] Suppose first that $f_c$ is an elementary expansion or elementary collapse, or the inverse of one. On cellular chains over $\mathbb Z[G]$, such a move changes the based chain complex by adjoining or deleting a two-term elementary acyclic summand \begin{align*} 0\to \mathbb Z[G]\xrightarrow{\operatorname{id}_{\mathbb Z[G]}}\mathbb Z[G]\to 0 \end{align*} up to elementary changes of basis. The torsion of this elementary summand is represented by the identity matrix and elementary basis changes; its class is therefore zero in the Whitehead quotient $\operatorname{Wh}(G)$. A cellular isomorphism sends each chosen lift of a cell to a lift of a cell multiplied by an element of $G$ and possibly by the sign from orientation. Thus its change-of-basis matrices represent products of units $\pm g$ and permutation matrices, and these have zero image in $\operatorname{Wh}(G)$ after quotienting by $\langle \pm g:g\in G\rangle$ and by elementary matrices. For a finite composite of elementary expansions, elementary collapses, cellular isomorphisms, and homotopies, the additivity theorem for Whitehead torsion under composition states that the torsion of the composite is the sum of the corresponding torsions, after transporting coefficients through the induced isomorphisms of fundamental groups. Each summand is zero by the preceding paragraphs, and homotopic cellular representatives have the same torsion by the convention in the statement. Hence every simple homotopy equivalence has torsion zero: \begin{align*} f \text{ simple }\implies \tau(f)=0. \end{align*} [guided] Let us spell out why elementary moves are invisible to Whitehead torsion. An elementary expansion attaches an $n$-cell together with an $(n+1)$-cell so that, in the universal cover, each lift of the new $(n+1)$-cell has boundary equal to the corresponding lift of the new $n$-cell plus terms already present in the old cellular chain complex. After subtracting those old terms by elementary basis changes, the new summand in cellular chains is the based two-term free $\mathbb Z[G]$-complex \begin{align*} 0\to \mathbb Z[G]\xrightarrow{\operatorname{id}_{\mathbb Z[G]}}\mathbb Z[G]\to 0. \end{align*} This complex is acyclic because the displayed boundary map is the identity. The contraction sends the lower-degree generator to the higher-degree generator and is represented by the identity matrix. The torsion matrix is therefore a product of identity matrices and elementary matrices, so its class maps to zero in $K_1(\mathbb Z[G])$ and hence to zero in $\operatorname{Wh}(G)$. A collapse is the inverse operation. It removes the same elementary acyclic summand, so it contributes the inverse torsion of a zero class; that inverse is again zero in $\operatorname{Wh}(G)$. A cellular isomorphism permutes cells, may reverse orientations, and may replace a chosen lift by its translate under an element of $G$. On the preferred cellular bases this gives permutation matrices and diagonal units $\pm g$, and the subgroup generated by those units is precisely quotiented out in $\operatorname{Wh}(G)=K_1(\mathbb Z[G])/\langle \pm g:g\in G\rangle$. Finally, the additivity theorem for Whitehead torsion under composition says that the torsion of a composite is the sum of the torsions of the factors after identifying coefficient groups. Thus a finite sequence of elementary expansions, elementary collapses, cellular isomorphisms, and homotopies has total torsion equal to a finite sum of zeros, hence has torsion zero. [/guided] [/step] [step:Use vanishing torsion to obtain a simple chain contraction] Assume now that \begin{align*} \tau(f)=0 \end{align*} in $\operatorname{Wh}(G)$. By the first step, we may work with the cellular representative $f_c$ and its lifted cellular chain map $C_*(\widetilde f_c)$. The mapping cone \begin{align*} D_*:=\operatorname{Cone}(C_*(\widetilde f_c)) \end{align*} is an acyclic based finite free $\mathbb Z[G]$-chain complex. The equality $\tau(f_c)=0$ says exactly that the torsion of $D_*$ is zero in $\operatorname{Wh}(G)$. We use the algebraic Whitehead criterion for finite based free chain complexes in the following precise form: an acyclic based finite free $\mathbb Z[G]$-chain complex has zero Whitehead torsion if and only if it is simply contractible. Here an elementary chain expansion means adjoining a two-term based summand $0\to \mathbb Z[G]\xrightarrow{\operatorname{id}_{\mathbb Z[G]}}\mathbb Z[G]\to0$ in adjacent degrees, an elementary chain collapse means deleting such a summand, and an elementary [change of basis](/page/Change%20Of%20Basis) means multiplying the preferred basis in one degree by an elementary matrix or by a signed group unit. The complex $D_*$ satisfies the hypotheses because it is acyclic, based by lifted cells, finite because $X$ and $Y$ are finite CW-complexes, and free over $\mathbb Z[G]$ because cellular chains of universal covers are free on lifted cells. Applying this criterion to $D_*$ shows that $D_*$ is simply contractible over $\mathbb Z[G]$. [guided] We now use the vanishing of the torsion class as an algebraic statement about the mapping cone. The complex \begin{align*} D_*:=\operatorname{Cone}(C_*(\widetilde f_c)) \end{align*} is a finite based free chain complex over $\mathbb Z[G]$: the basis in each degree is the preferred collection of lifted cells from $X$ and $Y$, and finiteness follows from the finiteness of the CW-complexes. It is acyclic because $\widetilde f_c$ is a homotopy equivalence of universal covers, so its cellular chain map is a chain homotopy equivalence and the cone of a chain homotopy equivalence is acyclic. The algebraic Whitehead criterion applies exactly to such objects. Its hypotheses are that the chain complex is acyclic, finite, based, and free over the ring in question. We have verified those hypotheses for $D_*$ over $\mathbb Z[G]$. The conclusion of the criterion is that zero torsion is equivalent to simple contractibility: $D_*$ can be reduced to the zero complex by finitely many elementary chain expansions, elementary chain collapses, and elementary changes of basis. An elementary chain expansion adjoins a two-term summand \begin{align*} 0\to \mathbb Z[G]\xrightarrow{\operatorname{id}_{\mathbb Z[G]}}\mathbb Z[G]\to0, \end{align*} an elementary chain collapse deletes such a summand, and an elementary basis change changes the preferred basis by elementary matrices or signed group units. Since $\tau(f_c)=0$ in $\operatorname{Wh}(G)$, the criterion gives that $D_*$ is simply contractible over $\mathbb Z[G]$. [/guided] [/step] [step:Realize the simple chain contraction by elementary CW moves] The simple contraction of \begin{align*} D_*=\operatorname{Cone}(C_*(\widetilde f_c)) \end{align*} is a finite algebraic sequence of elementary chain expansions, elementary chain collapses, and elementary basis changes over $\mathbb Z[G]$. By the cellular realization theorem for simple chain contractions, if $h:A\to B$ is a cellular homotopy equivalence of connected finite CW-complexes and the based $\mathbb Z[\pi_1(B)]$-cellular mapping cone of a lifted chain map is simply contractible, then $h$ is simple: the simple chain contraction is realized geometrically by finitely many elementary CW-expansions, elementary CW-collapses, and cellular isomorphisms. The map $f_c:X\to Y$ satisfies these hypotheses: it is cellular, it is a homotopy equivalence, $X$ and $Y$ are connected finite CW-complexes, and the preceding step proves that its lifted cellular mapping cone $D_*$ is simply contractible over $\mathbb Z[G]$. Therefore $f_c$ is a simple homotopy equivalence. Since $f$ is homotopic to $f_c$ and simple homotopy equivalence is invariant under replacing a map by a cellular homotopic representative in this convention, $f$ is a simple homotopy equivalence. This proves \begin{align*} \tau(f)=0\implies f \text{ is simple}. \end{align*} Combining this implication with the forward implication proves the equivalence. [/step]