A closed, oriented $d$-manifold $M$ is said to *dominate* another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of *homotopical domination* that Wall's finiteness obstruction relates to.)


This relation turns $d$-manifolds into a poset (not quite; the relation is not antisymmetric even if we identify diffeomorphic manifolds, as there are examples of manifolds that dominate each other and are not homotopy equivalent; but this subtlety is rather irrelevant) with $S^d$ as its smallest element. It has been of interest to geometric topologists at least since the times of Hopf to study this poset. For a survey on this topic, see <http://www.map.mpim-bonn.mpg.de/images/c/cf/Brouwer_degree.pdf>.


My question is: can *every* manifold $N$ be dominated by another manifold $M$ that is (stably) parallelizable, i.e., has trivial stable tangent bundle?



Can every manifold be dominated by a parallelizable one?
I think this follows from the stable Hurewicz map $\pi\_n^s(N) \to H\_n(N;\mathbb{Z})$ being an isomorphism modulo torsion. If we set $n = \dim(M)$ then some positive multiple of the fundamental class of $M$ is hit. Under the Pontryagin-Thom isomorphism this gives a stably framed $M$ and a continuous $f: M \to N$ such that $f\_\*([M])$ is a positive multiple of $[N]$.


