Given that $H_n(X)$ is free abelian, I'm trying to find the homology groups of $Y=S^1\times X$ using the Mayer-Vietoris theorem.
My first attempt decomposed $Y$ as $A \cup B$ where $A=\{*\}\times X$ and $B = S^1 \setminus \{*\}\times X$. Then $B$ clearly homotopy equivalent to $A$ and the Mayer-Vietoris sequence gives that $H_n(Y)=H_n(X)\bigoplus H_n(X)$.
This is clearly false (the torus is one easy counterexample). I assume that where I went wrong in is assuming that $B$ was triangulable. Am I right?
What would be the right way to go about this? Apart from what I tried I see no other decompositions which would work! I must be missing something.