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 $$g \ge h$$.