4
$\begingroup$

From Corollary 4.33 of Hatcher (which is the corollary of Hurewicz Theorem) the map $f:X \to Y$ between simply connected CW complexes is a homotopy equivalence if $f$ induces a quasi-isomorphism $f_{\ast}\colon H_{n}(X) \to H_{n}(Y)$.

Is there any example of two CW complexes (which are not simply connected) which are quasi-isomorphic but not homotopy equivalent?

$\endgroup$
  • $\begingroup$ Can't you just take $X=K(G,1)$ where $G$ is perfect and $Y$ the one point space? $\endgroup$ – Noel Lundström Mar 16 at 11:57
3
$\begingroup$

Let $M$ be an integral homology sphere which is not a sphere, i.e. a closed $n$-dimensional manifold with $H_*(M; \mathbb{Z}) \cong H_*(S^n; \mathbb{Z})$ but $\pi_1(M) \neq 0$. An example is the Poincaré dodecahedral sphere. Every closed orientable manifold admits a degree one map to a sphere of the same dimension, so there is a map $f : M \to S^n$ of degree one. The induced map on homology is an isomorphism, but $M$ is not homotopy equivalent to $S^n$ as $\pi_1(M) \neq 0$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thank you very much! $\endgroup$ – user124697 Mar 15 at 0:28
3
$\begingroup$

Quillen's plus construction produces examples of quasi-isomorphic CW-complexes that have different fundamental groups (by attaching 2-cells that change the fundamental group and then reversing the effect of these attachments on homology by attaching 3-cells).

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thank you very much for introducing the construction! $\endgroup$ – user124697 Mar 15 at 0:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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