Computing homology group of product of spheres

Question

I am having trouble computing $\tilde{H}_n(S^3\times S^1)$. I am supposed to use Mayer-Vietoris sequence. I know $\tilde{H}_n(A\vee B)\cong\tilde{H}_nA\oplus\tilde{H}_nB$ if there is a contractible neighborhood of the base point. But how to relate the product with the wedge sum if my direction is correct (Use wedge sum)?

Answer 1

Hint: write $S^3 \times S^1$ as the union of two copies of $S^3 \times I$.