4
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ It seems you have misquoted the paper slightly. They actually talk about an "asphericalization" $f_K:a(K)\to K$, so the map goes the other way. The claim that $f_K$ induces a surjection on any generalized homology theory is then supposed to follow from (2') $f_K$ injective on ordinary homology, and (4) $f_K$ pulls the stable tangent bundle of $K$ back to that of $a(K)$. $\endgroup$
    – Mark Grant
    Sep 19, 2018 at 6:17
  • $\begingroup$ @MarkGrant Yes , Thanks, I will edit it. $\endgroup$
    – BiM
    Sep 19, 2018 at 9:08

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.