TITLE: Are these two spaces homotopy equivalent?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: This is actually a special case of the result that if $A \to X$ is a closed cofibration, and $f:A \to B$ is a map, then the natural map $M(f) \cup X \to B \cup_f X$ is a homotopy equivalence: here $M(f)$ is the mapping cylinder of $f$, and the result is 7.5.4 of my book Topology and Groupoids. 
For your case, you have to take $A$ to consist of $4$ points, and $B$ to consist of $2$  points.  
Here is part of the general picture

and here is a picture of the special case you asked about but with just one pair of points identified: