Let $F:M\rightarrow N$ be a map between orientable compact connected $n$-manifolds such that $F^{-1}(\partial N)=\partial M$. The degree of $F$, $deg(F)$, is given by the equation $$F_{\#}([M])=deg(F)[N],$$where $F_{\#}:H_n(M,\partial M)\rightarrow H_n(N,\partial N)$ is the homomorphism induced by $F$ in the $n$-dimensional relative homology groups, $[M]\in H_n(M,\partial M)$ and $[N]\in H_n(N,\partial N)$ are the chosen fundamental classes of $M$ and $N$.
Assume that $deg(F)=1$. Is there any relation between the number of connected components of $\partial M$ and the number of connected components of $\partial N$?