I am trying to compute the cohomology of a particular space using the Mayer-Vietoris long exact sequence; I can't really figure out what the decomposition looks like. The space is the following:
Consider $S^3 \times S^3$ and remove a tube homeomorphic to $S^1 \times D^5$, now glue $D^2 \times S^4$ and call $N$ the resulting space. I have to show that $H^*(N; \mathbb{Z})$ is free with dimension $1, 0, 1, 2, 1, 0, 1$ on degree $i = 0, \ldots, 6$ respectively.
When I try to decompose $N$, I choose open sets $U,V$ such that $U \cong S^3 \times S^3$ and $V \cong D^2 \times S^4$, and then $U \cap V$ will be homeomorphic to the common boundary $S^1 \times S^4$ along the gluing ; however, with this decomposition, I am not able to get the final answer, my problem arises in degree $4$; to get the given answer, I will need $U \cap V \cong S^1$ which I exactly do not how to see.