# Do "surjective" degree zero maps exist?

Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not homotopic to a non-surjective map?

Added: The motivation is: There is a "mild version" of the Nearby Langrangian conjecture stating: any exact Lagrangian manifold $X \to T^*Y$ has non-zero degree when composed with the projection $T^*Y \to Y$. It is known that the map is always surjective. I am looking at a possible inbetween stating that the map cannot be homotoped to a non-surjective map.

• I seems like it will be very very hard to prove that a given map is not homotopic to a non-surjective map. Apr 8 '10 at 12:42
• Partial answer: If $Y=S^n$, it follows by the Theorem of Hopf that the degree determines the homotopy class. It's on the last page before the exercises in Milnor's Topology from a Differentiable Viewpoint. This gives a negative answer for spheres, but I don't know about the general case. Also, by closed, do you mean closed as a submanifold of euclidean space? Apr 8 '10 at 13:44
• "Closed" is standard terminology for a compact manifold without boundary. Apr 8 '10 at 13:57
• Ah, I've never heard of that before. Apr 8 '10 at 14:42
• @Bogdan: You don't need to remove smoothness for that, and that is not the question. You may have missed the part "homotopic to". Jul 22 '14 at 12:02