Let $f:M\rightarrow W$ be a smooth map between smooth manifolds such that both $M$ and $W$ are compact, connected and oriented. Let $N\subseteq W$ be a closed submanifold. Let us suppose that $$ \#(f,N)=0. $$ Does a smooth map $g:M\rightarrow W$ homotopic to $f$ and such that $g(M)\cap N=\varnothing$ exist?

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    $\begingroup$ I first thought it should be an immediate "generalization" of the usual Hopf degree theorem argument that a vector bundle of rank $k$ with zero Euler characteristic on a $k$-dimensional manifold has a nowhere vanishing section. But it is — not too surprisingly — hard to make an argument work. My topology friends in MSE chat told me quickly that it's false. Try two curves on a genus 2 torus — one chopping the torus in half vertically, the other a circle going from the edge of one hole to the edge of the other and back. $\endgroup$ – Ted Shifrin Nov 8 '17 at 22:32
  • $\begingroup$ Are you familiar with the fundamental group and its relation to the 1st homology? If so, you can construct a counter-example to your conjecture by taking $M=S^1$ and $W$ the genus 2 surface. $\endgroup$ – Moishe Kohan Nov 9 '17 at 3:07

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