= Homology of an integer torus endomorphism
{title2=$(f_A)_*|_{H_q}=\bigwedge^q A$}
On the <torus> $\mathbb R^d/\mathbb Z^d$, an integer <matrix> $A$ defines an <endomorphism>. The coordinate circles give a basis of the <first homology group>, and the induced map on that group is $A$. The <Künneth theorem> identifies higher <integral homology> with the <exterior powers> of $\mathbb Z^d$, so the degree-$q$ map is $\bigwedge^q A$. In top degree it is multiplication by $\det A$. The map is a <homeomorphism> precisely when $A$ is a <unimodular matrix>; a merely nonsingular integer <matrix> need not have an integer inverse.
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