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

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) C1 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 ϕ. In such a case, we call ϕ the suspension of h.