There are indeed counterexamples to which Igor Belegradek gave reference. Here is aanother counterexample in the plane, perhaps the simplest there is: Let $X$ be an annulus with two disjoint arcsone arc attached to one of its boundary components and another arc attached to the other boundary component, and $Y$ - an an annulus with one arctwo disjoint arcs attached to the same one of its boundary components and another arc attached to the other boundary component.
Here is a counterexample in the plane: Let $X$ be an annulus with two disjoint arcs attached to one of its boundary components, and $Y$ - an annulus with one arc attached to one of its boundary components and another arc attached to the other boundary component.
There are indeed counterexamples to which Igor Belegradek gave reference. Here is another counterexample in the plane, perhaps the simplest there is: Let $X$ be an annulus with one arc attached to one of its boundary components and another arc attached to the other boundary component, and $Y$ - an annulus with two disjoint arcs attached to the same one of its boundary components.
