Homology of product of spheres

Statement

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

$b_{q}(A)=\left|\left\{T\subset \{1,2,3,\ldots ,r\}|\sum _{i\in T}m_{i}=q\right\}\right|$

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

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

$b_{q}(A)={r \choose q}$

An alternative interpretation of the above result is that $b_{q}(A)$ is the coefficient of $x^{q}$ in the product:

$\prod _{i=1}^{r}(1+x^{m_{i}})$

Related invariants

Euler characteristic

The Euler characteristic of the product of spheres can be obtained by plugging $(-1)$ in the above polynomial. From this it turns out that the Euler characteristic is $0$ if any of the spheres has odd dimension, and is $2^{r}$ if all the spheres have even dimension.

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.