The Poincaré-Bendixson theorem for the Klein bottle

Abstract

we see that ao is a group of homeomorphisms under the usual composition of maps. We define a to be the group of homeomorphisms generated by To01 and K where K(z) -+ 1/2. Then C/$o and C/$ are the torus Y and the Klein bottle X, respectively. In addition, (C, p) and (C, p') are the universal covering spaces of 5r and X where p and p' are the canonical maps. Because ao ' a we get a natural map P2: Y-> X such that (g, P2) is the two-fold regular covering space of Xy and p' =P2 ? PLet (X, p) be a covering space of X and let (X, R, r) be a continuous flow. Then there exists a unique flow (X, R, r such that p is a homomorphism of (X, R, rr) onto (X, R, r) [2]. Moreover, x E X is a fixed point of X if and only if p(x) is a fixed point of X and the covering transformations are automorphisms of (X, R, r). Let (X, R, r) be a continuous flow on a two-manifold and let x E X. A local cross section of X at x is a subset S of X containing x which is homeomorphic to a nondegenerate closed interval and for which there exists an ? > 0 such that the map (s, t) ->v(s, t) is a homeomorphism of S x [s, s] onto the closure of an open neighborhood of x. We call s the length of the local cross section. If x is an interior point of X which is not a fixed point of v, then there exists a local cross section of X at x [4]. Let S be a simple curve and a, b E S. We denote the open segment of S between a and b by (a, b)s. For r>0 and a e X, (a, ar)={v(a, t): t e (0, r)} and [a, ar] ={I(a, t) : t E [0, r]}, where X is a continuous flow on X. Let ls (z, z') and 1 (z, z')