MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
I seems like it will be very very hard to prove that a given map is not homotopic to a non-surjective map. – Chris Schommer-Pries 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? – Harry Gindi Apr 8 '10 at 13:44
"Closed" is standard terminology for a compact manifold without boundary. – Tyler Lawson Apr 8 '10 at 13:57
Ah, I've never heard of that before. – Harry Gindi Apr 8 '10 at 14:42
If you're getting into this "nearby Lagrangian" stuff, make sure you're up to date! You need to know the theorem of Fukaya-Seidel-Smith/Nadler about Maslov-zero exact Lagrangians in simply connected cotangent bundles, and the recent work of Abouzaid about cotangent fibres, relevant to relaxing the simple connectedness assumption. – Tim Perutz Apr 8 '10 at 16:29
up vote 32 down vote accepted

It is a theorem of H. Hopf that a map between connected, closed, orientable n-manifolds of degree 0 is homotopic to a map that misses a point, when n > 2. See D. B. A. Epstein, The degree of a map. Proc. London Math. Soc. (3) 16 1966 369--383, for a "modern" discussion including the analogous situation in the non-orientable case. The same result holds for n = 2, but is more difficult and is due to Kneser. See Richard Skora, The degree of a map between surfaces. Math. Ann. 276 (1987), no. 3, 415--423, for a thorough discussion of the non-orientable case in dimension 2.

share|cite|improve this answer
Great - Thanks! I am assuming that you forgot the connected assumption. – Thomas Kragh Apr 8 '10 at 20:03
Right. "All manifolds are connected unless otherwise stated." I'll fix it for the record. – Allan Edmonds Apr 9 '10 at 1:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.