Isomorphism in cohomology

Question

Let $M$ be a manifold with boundary $\partial M=N_1 \cup N_2$, $N_i$ are connected manifolds. If: 

$$H^*(M,\partial M)=0$$

then, have we:

$$H^*(N_1,{\bf R})\cong H^*(N_2,{\bf R})$$

?