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Let $X$ be the $2$-sphere with two pairs of points identified, say $(1,0,0) \sim (-1,0,0)$ and $(0,1,0) \sim (0,-1,0)$. Write $Y$ for the wedge sum of two circles with a $2$-sphere: if it matters, the sphere is in the "middle," so the circles are attached at two distinct points on the sphere.

Now I think one can show, using Mayer-Vietoris and van Kampen, that these spaces have the same homology (that of a torus) and fundamental group (free on two generators). But are they homotopy equivalent?

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    the cup product structure on the cohomology ring of $Y$ is trivial; I doubt this is true on $X$ - you could check with a simplicial decomposition; I'll try to think of a less awful way.2012-08-07
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    Hint: $X$ and $Y$ have the same cohomology ring, the same homotopy groups, homotopy algebras, $K$-theory, etc, etc...2012-08-07

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