Surjections on generalized homology theory

Question

In Davis-Januszkiewica´s paper Hyperbolization of polyhedra , the authors hyperbolized every closed n-manifolds K to get a new manifold, say M(K),together with a map $f_K$ from M(K) to K, then they claimed that $f_K$ induces a surjection on any generalized homology theory.(see Theorem B in the introduction)

According to the authors, this claim is the result of combining two facts:

1. For any homology with local coefficients, $f_K$ induced an injection.

2. $f_K$ pulls back the stable tangent bundle of M(K) to the stable tangent bundle of K.

I don´t know how to deduce the claim from these two facts. Without backgrounds on generalized homology theory, I only know some basic definitions such as stable tangent bundle, so I don´t understand how these two facts can be used in generalized homology theory.

Can anyone gives some details? Thank you.

Answer 1

I would guess that (2') has a typo, and "into" should be "onto".

Otherwise the statement is of course not true: the degree $d$ map $S^2 \to S^2$ is injective on homology with all local coefficients, and pulls back the stable tangent bundle of $S^2$ to that of $S^2$, as both are trivial. But it of course not surjective on ordinary homology.