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 $N$ .

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 $n^{t h}$ homology.
In general, if $M$ and $N$ are compact connected orientable manifolds, then there exist degree one maps from $M ♯ N$ 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 $M ♯ S^{n}$ .