Cohomology groups do not determine homotopy type
= Cohomology groups do not determine homotopy type
The <torus> and the <wedge sum> $S^1\vee S^1\vee S^2$ both have integral cohomology $\mathbb Z,\mathbb Z^2,\mathbb Z$ in degrees zero, one and two. Their fundamental groups differ: the torus has $\mathbb Z^2$, while the wedge has the nonabelian free group on two generators. Thus even compact finite cell complexes can have the same additive cohomology without being homotopy equivalent.