Homology of quaternionic projective space

Statement

The homology of quaternionic projective space is given as follows:

$H_p(\mathbb{H}{P}^n)=\mathbb{Z}p=0,4,8,\dots ,4n$

and is zero otherwise.

Related invariants

The Betti numbers of quaternionic projective space are thus $1$ for $4k$ with $0\le k\le n$ and $0$ elsewhere. Thus, the Euler characteristic is $n+1$.

Proof

We use the cell decomposition of quaternionic projective space with one cell each in dimensions $0,4,8,\dots$. The cellular chain groups are thus $\mathbb{Z}$ in positions $0,4,\dots ,4n$ and 0 elsewhere. This forces the cellular homology groups to also be $\mathbb{Z}$ exactly in those positions.