Example of cw complexes quasi isomorphic but not homotopic.

Question

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?

Answer 1

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 $\pi_1(M) \neq 0$.

Answer 2

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).