Using simplicial volume to count multi-tangent trajectories of traversing vector fields

作者：Alpert Hannah^*; Katz Gabriel

来源：Geometriae Dedicata, 2016, 180(1): 323-338.

DOI：10.1007/s10711-015-0104-6

• Traversing vector field
• Simplicial volume
• Simplicial norm
• Amenable group

For a non-vanishing gradient-like vector field on a compact manifold Xn+1 with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity n, which is the maximum possible. (Among them are trajectories that are tangent to partial derivative X exactly n times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of X by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds.

• 出版日期2016-2