Degree does not classify general manifold maps
= Degree does not classify general manifold maps
Although <topological degree> classifies maps $S^n\to S^n$ up to <homotopy>, it is not a complete invariant for other manifolds. On the <torus>, the identity and the integer shear $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ both have degree one, but induce distinct maps on $H_1(T^2;\mathbb Z)$ and cannot be homotopic.