Integral cohomology of an inversion mapping torus
= Integral cohomology of an inversion mapping torus
For $r=n-1$, the <Wang sequence> gives an additive splitting
$$
H^q(M_n;\mathbb Z)\cong
\operatorname{coker}((-1)^{q-1}-1:\mathbb Z^{\binom r{q-1}}\to\mathbb Z^{\binom r{q-1}})
\oplus
\ker((-1)^q-1:\mathbb Z^{\binom rq}\to\mathbb Z^{\binom rq}).
$$
The kernel is free, so the splitting exists as groups. The map in even degree is zero; the map in odd degree is multiplication by minus two.