 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 $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ both have degree one, but induce distinct maps on $H_1(T^2;\mathbb{Z})$ and cannot be homotopic.