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.