Example

Consider the product of spheres $S_2 \times S_2$ and the complex projective space $P_2(\mathbf{C})$. Let $b \in S_2$, we define a cellular decomposition $(Y_n)$ of $Y = S_2 \times S_2$ by setting

this decomposition has one cell of dimension 0, two of dimension 2 and one of dimension 4. The differentials of the associated complex are necessarily zero, so $H_0(Y)$, $H_2(Y)$ and $H_4(Y)$ are free of rank 1, 2 and 1 respectively, and $H_n(Y) = 0$ for $n \notin \{ 0, 2, 4 \}$. One obtains a cellular decomposition $(Z_n)$ of $P_2(\mathbf{C})$ by setting

the space $P_1(\mathbf{C})$ being embedded in $P_2(\mathbf{C})$ (TG, VIII, p. 20), and c being a point of $P_1(\mathbf{C})$; this decomposition has one cell of dimension 0, one of dimension 2 and one of dimension 4. Here again the differentials of the complex are necessarily zero, and it follows that $H_n(P_2(\mathbf{C}))$ is isomorphic to A for $n \in \{ 0, 2, 4 \}$ and to 0 otherwise. Since the homology modules in degree 2 of the two spaces considered are free of rank 2 and 1 respectively, these spaces are not homeomorphic. \*