Every nonsingular C1 flow on a closed manifold of dimension greater than two has a global transverse disk

Abstract

We prove three results about global cross sections which are disks, henceforth called global transverse disks. First we prove that every nonsingular (fixed point free) C^1 flow on a closed (compact, no boundary) connected manifold of dimension greater than 2 has a global transverse disk. Next we prove that for any such flow, if the directed graph Gh has a loop then the flow does not have a closed manifold which is a global cross section. This property of Gh is easy to read off from the first return map for the global transverse disk. Lastly, we give criteria for an ''M-cellwise continuous'' (a special case of piecewise continuous) map h:D2->D2 that determines whether h is the first return map for some global transverse disk of some flow phi. In such a case, we call phi the suspension of h.