Let $X$ and $Y$ be connected finite CW-complexes, let $f:X\to Y$ be a homotopy equivalence, and choose a basepoint $x_0\in X$. Put $y_0=f(x_0)$ and $G:=\pi_1(Y,y_0)$, and use the isomorphism $f_*:\pi_1(X,x_0)\to G$ to regard cellular chains of the universal cover of $X$ as left $\mathbb Z[G]$-modules. Choose a cellular representative $f_c:X\to Y$ homotopic to $f$, universal covers of $X$ and $Y$, compatible lifts of cells giving preferred bases, and a lift of $f_c$. Let $\tau(f)\in \operatorname{Wh}(G)$ be the Whitehead torsion class of the resulting based finite free $\mathbb Z[G]$-cellular mapping cone, with the standard convention that this class is independent of the cellular representative, the lift, the preferred lifted-cell bases up to multiplication by units $\pm g$, and cellular homotopy. Here the Whitehead group is

\begin{align*} \operatorname{Wh}(G):=K_1(\mathbb Z[G]) / \langle \pm g : g\in G\rangle. \end{align*}

Then $f$ is a simple homotopy equivalence, meaning homotopic to a cellular map obtained by a finite sequence of elementary CW-expansions, elementary CW-collapses, and cellular isomorphisms, if and only if

\begin{align*} \tau(f)=0 \end{align*}

in $\operatorname{Wh}(G)$.