Degree one map

Definition
Let $$M$$ and $$N$$ be compact connected orientable manifolds. Choose fundamental classes for $$M$$ and $$N$$. Then a degree one map from $$M$$ to $$N$$ is a continuous map $$f:M \to N$$ such that $$f$$ sends the fundamental class of $$M$$ to the fundamental class of $$N$$.

Facts

 * There always exists a degree one map from any compact connected orientable manifold, to the sphere. The map is constructed as follows: let $$M$$ be the compact connected orientable manifold, and $$p \in M$$ be a point. Suppose $$D$$ is a closed disc containing $$p$$ inside a Euclidean neighbourhood of $$p$$. Let $$A$$ denote the complement of $$D$$. The map $$M \to M/A = S^n$$ is a degree one map, viz it induces an isomorphism on the $$n^{th}$$ homology.
 * In general, if $$M$$ and $$N$$ are compact connected orientable manifolds, then there exist degree one maps from $$M \sharp N$$ to $$M$$ and to $$N$$. The map to $$M$$, for instance, pinches the entire part from $$N$$ to a point. The previous fact can be viewed as a special case of this, by viewing $$M$$ as a connected sum $$M \sharp S^n$$.
 * Any degree one map induces a surjective map on fundamental groups, and hence in particular on the first homology group
 * Any degree one map induces a surjective map on all homology groups