Whitehead torsion is the complete obstruction to finite CW simple homotopy

If $f$ is simple, [F1] gives $\tau (f)=0$ on each target component.

Conversely suppose $\tau (f)=0$. By [F7] and [F4] replace $f$ by a cellular map in its homotopy class; this changes neither its torsion nor whether it is simple. The construction is finite because $X$ is finite. Work first on one connected component; a homotopy equivalence bijects the finite component sets.

Form the finite cellular mapping cylinder $M_f$. Its target inclusion $j:Y\hookrightarrow M_f$ is simple by [F2], so [F1] gives $\tau (j)=0$. Since $p\circ j={\mathrm{id}}_Y$, [F5] yields $0=\tau (p)+p_{\ast }\tau (j)=\tau (p)$. Also $f=p\circ i_X$, whence $0=\tau (f)=\tau (p)+p_{\ast }\tau (i_X)=p_{\ast }\tau (i_X)$. The retraction $p$ is a homotopy equivalence and induces an isomorphism on Whitehead groups, so $\tau (i_X)=0$.

The source inclusion $i_X:X\hookrightarrow M_f$ is a homotopy equivalence of finite connected CW complexes. Apply [F3] to obtain a finite relative elementary deformation from $M_f$ to $X$. Reversing that sequence shows $i_X$ is simple. The target inclusion $j$ is also simple, so its inverse retraction $p$ is homotopic to the reverse composite of its elementary moves and is simple by [F6]. Thus $f=p\circ i_X$ is simple. Repeat on each of the finitely many components and concatenate their finite move sequences. This proves the reverse implication and the asserted direct-sum statement. ∎