The unitary group U(N) <br>
has the same cohomology ring <br>
(and dimension) as the product of spheres <br>
S^1 x S^3 x ... S^{2N-1} <br>
but the two spaces have different <br>
homotopy types as soon as  N>2.

[see Hatcher, Algebraic Topology, p. 434]