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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 $N$. Consider $S$ and $\Sigma$ orientable compact connected hypersurfaces in $M$ and $N$, respectively, such that $\partial S=S\cap \partial M$ and $\partial\Sigma=\Sigma\cap \partial N$.

Is it true that, if $deg(F)=1$, $F(S)=\Sigma$ and $F^{-1}(\partial\Sigma)=\partial S$, then $deg(f)=1$, where $f=\left.F\right|_S$?

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This seems false to me, as stated. To convince yourself, take $M=N=S^1$ (so we don't have to worry about boundaries). Take $F$ of degree $1$ but with a regular value $y$ having $3$ preimage points $x,x',x''$, at which $F$ has local degree $-1$, $1$, and $1$ respectively. (You can draw this easily by mapping $S^1\to\Bbb R^2-\{0\}\to S^1$.) Now take $S=\{x\}$ and $\Sigma =\{y\}$.

(You can suspend to higher dimensions and even set it up so that, for example, $M$ and $N$ are closed hemispheres.)

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