Integer-matrix torus covering

A nonsingular integer matrix $A$ induces the torus covering

p_A:\mathbb T^n\longrightarrow\mathbb T^n,\qquad [x]\longmapsto[Ax].

Integer entries make the map well-defined. It is a covering map with $|\det A|$ sheets.

Fibers and local inverses

The map is onto since $A^{-1}y$ is a preimage of $y$ . Its kernel is $A^{-1}\mathbb Z^n/\mathbb Z^n$ , identified by multiplication by $A$ with $\mathbb Z^n/A\mathbb Z^n$ . Every fiber is a translate of this finite kernel. Small neighborhoods of the finitely many preimages give disjoint inverse branches of the linear map, proving the covering property. If $|\det A|=1$ , the map is a torus automorphism. If $|\det A|>1$ , it is not injective; local lifts must retain their sheet labels.

References

Hatcher, Algebraic Topology, §1.3 (covering spaces and deck transformations).