I'm trying to understand the solution of the following problem ($S^1\times D^2/S^1\times S^1\approx S^2\vee S^3$ - homotopy equivalence) : Prove homotopy equivalence of two spaces .
Implicitly there was used a fact that $(S^1\times D^2)\cup_{S^1\times S^1}(S^1\times D^2)=\partial D^4=S^3$ and I don't know how to prove it. Is this more general fact that $(A\times B)\cup_{A\times A}(A\times B)=\partial(B\times B)$ or some other generalization?
I couldn't add a comment, I hope it's ok to make a post like that. Thanks!