Do "surjective" degree zero maps exist? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:37:23Z http://mathoverflow.net/feeds/question/20712 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20712/do-surjective-degree-zero-maps-exist Do "surjective" degree zero maps exist? Thomas Kragh 2010-04-08T12:11:50Z 2010-04-09T01:25:25Z <p>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 <strong>homotopic</strong> to a non-surjective map?</p> <p><strong>Added</strong>: 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 <strong>possible</strong> inbetween stating that the map cannot be homotoped to a non-surjective map.</p> http://mathoverflow.net/questions/20712/do-surjective-degree-zero-maps-exist/20734#20734 Answer by Paul for Do "surjective" degree zero maps exist? Paul 2010-04-08T14:35:43Z 2010-04-08T15:06:22Z <p>deleted wrong answer</p> http://mathoverflow.net/questions/20712/do-surjective-degree-zero-maps-exist/20759#20759 Answer by Allan Edmonds for Do "surjective" degree zero maps exist? Allan Edmonds 2010-04-08T16:58:07Z 2010-04-09T01:25:25Z <p>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.</p>