Degree one map

This article defines a property of a continuous map between compact connected orientable manifolds (the property may strictly require a choice of orientation)

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 $M\mathrm{♯}N$ 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 $M\mathrm{♯}{S}^n$.
• 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