Let $M$ be a compact manifold. I want to understand why $$ \beta_i(M\times M, M) = \beta_i(M\times M)-\beta_i(M), $$ where $\beta_i$ denotes the $i$th Betti number and we identify $M$ with the diagonal in $M\times M$.
Taking a tubular neighborhood $U\subset M\times M$ of $M$, we define a manifold with boundary by $N:= M\times M - U$. By excision we have $$ H_i(N,\partial N) \cong H_i(M\times M, M). $$ Here I am stuck. Is this the right approach? If so, how can one compute $H_i(N,\partial N)$? If not, what is?