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?
Best Answer
Yes. Taking the wedge sum with a circle is the same as identifying two points (up to homotopy, with a nice space like the sphere which is homogeneous).