Quasiorder on compact connected orientable manifolds

Definition

Consider the class $C$ of all $n$-dimensional compact connected orientable manifolds. Then for any ring of coefficients $R$, we get a quasiorder on $C$ (dependent on $R$), as follows: we say that $M\ge N$ if there is a continuous map from $M$ to $N$ that induces an isomorphism on the $n^{th}$ homology.

When the ring $R=\mathbb{Z}$, this is equivalent to demanding that, for suitable orientations of $M$ and $N$, there is a degree one map from $M$ to $N$.

Facts

• A connected sum is always higher in the quasiorder than each of the summands, whatever the ring of coefficients.
• The $n$-sphere is the lowest in the quasiorder -- there is always a degree one map from any compact connected orientable manifold to the $n$-sphere

The quasiorder in two dimensions

In two dimensions, the quasiorder is in fact a total order on the manifolds upto homeomorphism: the lowest is the sphere. The surface of genus $g$ is higher in the quasiorder than the surface of genus $h$ iff $g\ge h$.