# Example of cw complexes quasi isomorphic but not homotopic.

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?

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

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 $$\pi_1(M) \neq 0$$.