Exist simply connected CW complexes $X$, $Y$ and a mapping $f:X\to Y$ with the property that the reduced suspension $\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence but $f$ is not?



Is a map a homotopy equivalence if its suspension is so?
Whitehead's Theorem (it is Corollary 4.33 in Allen Hatcher's [book](http://www.math.cornell.edu/~hatcher/#ATI)) says that a map between simply connected CW-complexes is a homotopy equivalence if and only if the induced map on homology (with $\mathbb Z$-coefficients) is an isomorphism. If $\Sigma f : \Sigma X \to \Sigma Y$ is a homotopy equivalence, then this is clearly the case, since suspension just shifts dimensions and the spaces are connected, so that there is no problem in dimension $0$.


It would be interesting to have an argument which does not use all the machinery that goes into Whitehead's Theorem, since your assumption is rather strong.


