Flows with cross sections.
Let M be a compact connected C(infinity)-manifold, of dimension n, without boundary. Let f(t): M --> M be a C(r)-flow with cross section. Let D(r)(M) be the topological group of diffeomorphisms of M with C(r)-topology (1 </= r </= infinity) and let D(o) (r)(M) be its connected component of the identity. Let [unk](M) be the group of I-cobordism classes in D(r)(M) generated by orientation-preserving diffeomorphisms. For fepsilonD(r)(M) denote by [f] its I-cobordism class. Theorem 1 deals with the dependence of M(f) on [f]. Theorem 2: S(6) x S(1) has at least 28 distinct differentiable structures.Let x(o)epsilonS(1) and let [unk](r) be the set of C(r)-flows (r >/= 1) in M x S(1) with cross section M x {x(o)} and inducing in it the identity. Theorem 3: Intuitively to a loop in D(o) (r) based at the identity there corresponds a flow in [unk](r), and to homotopic loops correspond isotopic flows.COROLLARY. complete analysis of [unk](r)/ [unk] for dim M = 2.Theorems 4 and 5 refer to Anosov flows for dim M > 3.
Verjovsky
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DIVISION OF APPLIED MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RHODE ISLAND.