Homology of products of spheres

Example 3.3.15. For \(n, m \geq 1\) with \(n \neq m\), the product \(S^n \times S^m\) has a CW-structure with one \(0\)-cell, one \(n\)-cell, one \(m\)-cell, and one \((n+m)\)-cell. If \(n,m\geq 2\) and \(|n-m|>1\), then no two cells are in adjacent dimensions, so \[ H_k(S^n \times S^m) \;\cong \; \begin {cases} \Z & \text {if } k = 0, n, m, \text { or } n+m, \\ 0 & \text {otherwise.} \end {cases} \] In the remaining cases, some cells lie in adjacent dimensions, but the corresponding cellular boundary maps are zero, giving the same answer.