Homology of product of spheres

Statement

Let $(m_{1}, m_{2}, \dots, m_{r})$ be a tuple of nonnegative integers. Let $A$ be the space $S^{m_{1}} \times S^{m_{2}} \times S^{m_{3}} \times \dots \times S^{m_{r}}$ . Then the homologies of $A$ are free Abelian, and the $q^{t h}$ Betti number is given by the following formula:

$b_{q} (A) = | {T \subset {1, 2, 3, \dots, r} | \sum_{i \in T} m_{i} = q} |$

In other words $b_{q} (A)$ is the number of ways $q$ can be obtained by summing up subsets of $(m_{1}, m_{2}, \dots, m_{r})$ .

A particular case of this is when all the $m_{i}$ s are 1. In this case:

$b_{q} (A) = (\binom{r}{q})$ .

Proof

Using exact sequence for join and product

Further information: exact sequence for join and product

The above claim can be easily proved using induction, and the exact sequence for join and product.

Statement

Proof

Using exact sequence for join and product

Using a CW-decomposition