Hopf conjecture

In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.

There are analogue conjectures if the curvature is allowed to become zero too. The statement should still be attributed to Hopf (for example in a talk given in 1953 in Italy). A compact, even dimensional Riemannian manifold with non negative sectional curvature has non negative Euler characteristic. A compact, (2d) dimensional Riemannian manifold with non positive sectional curvature has Euler characteristic of sign or zero. This version was stated as such as Question 1 in the paper or then in a paper of Chern. An example for which the conjecture is confirmed is for the product of 2 dimensional manifolds with curvature sign As the Euler characteristic satisfies which has the sign , the sign conjecture is confirmed in that case (if for all k, then and if for all k, then for even d and for odd d, and if one of the is zero, then ).

Hopf asked whether every continuous self map of an oriented closed manifold of degree 1 is necessarily a homotopy equivalence. It is easy to see that any map of degree 1 induces a surjection on ; if not, then factors through a non trivial covering space, contradicting the degree 1 assumption. This implies that the conjecture holds for Hopfian groups, as for them one then gets that is an isomorphism on and thus a homotopy equivalence. There are, however, some non Hopfian groups.

There is a conjecture which relates to the Hopf sign conjecture but which does not refer to Riemannian geometry at all. Aspherical manifolds are connected manifolds for which all higher homotopy groups disappear. The Euler characteristic then should satisfy the same condition as a negatively curved manifold is conjectured to satisfy in Riemannian geometry:

Suppose M2k is a closed, aspherical manifold of even dimension. Then its Euler characteristic satisfies the inequality

There can not be a direct relation to the Riemannian case as there are aspherical manifolds that are not homeomorphic to a smooth Riemannian manifold with negative sectional curvature. This topological version of Hopf conjecture is due to William Thurston. Ruth Charney and Michael Davis conjectured that the same inequality holds for a non positively curved piecewise Euclidean (PE) manifold.

There had been a bit of confusion about the word "Hopf conjecture" as an unrelated mathematician Eberhard Hopf and contemporary of Heinz Hopf worked on topics like geodesic flows. (Eberhard Hopf and Heinz Hopf are unrelated and might never have met even so they were both students of Erhard Schmidt). There is a theorem of Eberhard Hopf stating that if the 2 torus has no conjugate points, then it must be flat (the Gauss curvature is zero everywhere). The theorem of Eberhard Hopf generalized a theorem of Marston Morse and Gustav Hedlund (a PhD student of Morse) from a year earlier. The problem to generalize this to higher dimensions was for some time known as the Hopf conjecture too. In any case, this is now a theorem: A Riemannian metric without conjugate points on the n dimensional torus is flat.