In problem 8-11 of Lee's "Intro to Topological Manifolds", the reader is asked to show that for every continuous $\phi:\mathbb{T}^2\to\mathbb{T}^2$, there is a $2\times 2$ integer matrix $D(\phi)$ with the following properties:
- $\phi,\psi$ are homotopic iff. $D(\phi)=D(\psi)$
- $D(\psi\circ\phi)=D(\psi)D(\phi)$
- For every $2\times 2$ integer matrix $E$, there is a continuous $\phi$ with $D(\phi)=E$
- $\phi$ is homotopic to a homeomorphism iff. $D(\phi)\in GL(2,\mathbb{Z})$
This is clearly some analogue of the degree of a map, as introduced in the book for maps from the circle to itself. Since $\mathbb{T}^2=S^1\times S^1$, the fundamental group is $\mathbb{Z}\times\mathbb{Z}$ and so endomorphisms of $\pi_1(\mathbb{T}^2)$ are indeed $2\times 2$ integer matrices. I have shown, without constructing such a matrix explicitly, that they satisfy 1) and 2). For 3), however, I am not sure how to do this without explicitly constructing the matrix for a given map $\phi$, and I am unsure how to construct this matrix. It is clearly related to the winding number of the map around the torus, but from that I can only extract two integers, as far as I can tell.
How would I construct this matrix, or alternatively, how could I prove that every endomorphism has at least one associated map?