Path fields on manifolds
Robert F. Brown · Transactions of the American Mathematical Society · 1965
Nash [11] defined a set T0 consisting of paths a on a topological manifold M such that for 0 ^ t S 1, a(0 = a(0) if and only if t = 0. We add to this set the constant paths on M; call the union T and give it the compact-open topology.The map q:T -> M defined by q(a) = <x(0) is a fibre space in the sense of Hurewicz [10].This fibre space extends the concept of the tangent bundle to topological manifolds in the sense that when M is a differentiable manifold, (T,q,M) is fibre homotopy equivalent to the tangent bundle of M under a fibre homotopy equivalence which takes T0 into the nonzero tangent vectors [6].We recall that a vector field on a differentiable manifold may be defined as a cross-section in the tangent bundle.By analogy, we define a path field on a topological manifold to be a crosssection in (T,q,M).The principal result on the existence of nonzero vector fields on differentiable manifolds was discovered by Hopf [9].The theorem states that a compact orientable differentiable manifold M admits a vector field which is nowhere the zero vector if and only if the Euler characteristic of M is zero.The purpose of this paper is to prove the corresponding theorem for path fields on topological manifolds: a necessary and sufficient condition for a compact orientable topological manifold to admit a path field which is nowhere the constant path is that the Euler characteristic of the manifold be zero.A somewhat modified proof of the existence theorem, developed by Alexandroff and Hopf [1], is the model for the present proof.The main step in their development is the proof that every compact differentiable manifold admits a vector field which is the zero vector at one point at most.In other words, there is always a cross-section in the tangent bundle which intersects the standard cross-section of zero vectors at one point at most.This theorem, besides being of interest in its own right, reduces the problem from a global to a local one which is much easier Presented to the Society, August 29,