de Rham cohomology of a double

Question

Recall a double of a manifold (with boundary) M is two copies of M with boundary glued together. Let me call it D. How can I calculate $H^*(D)$ from $H^*(M)$?

I know a trivial example is the double of a half plane (which is The Whole plane). What about some more complicated examples?

Answer 1

Consider $M_1$ and $M_2$ the two copies, and $U$ a tubular neighborhood of $\partial M$ in $D$, then apply Mayer-Vietoris to $A=M_1 \cup U$, $B=M_2 \cup U$, giving an exact long sequence $H^i(D) \rightarrow H^i(M_1 \cup U) \oplus H^i(M_2 \cup U)=H^i(M)^2 \rightarrow H^i(U)=H^i(\partial M) \rightarrow H^{i-1}(D)$.