Asymptotic homotopy cycles for flows and Π₁ de Rham theory

Benardete, Diego; Mitchell, John

doi:10.1090/S0002-9947-1993-1093216-6

Asymptotic homotopy cycles for flows and $\Pi _ 1$ de Rham theory
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by Diego Benardete and John Mitchell PDF

Trans. Amer. Math. Soc. 338 (1993), 495-535 Request permission

Abstract:

We define the asymptotic homotopy of trajectories of flows on closed manifolds. These homotopy cycles take values in the $2$-step nilpotent Lie group which is associated to the fundamental group by means of Malcev completion. The cycles are an asymptotic limit along the orbit of the product integral of a Lie algebra valued $1$-form. Propositions 5.1-5.7 show how the formal properties of our theory parallel the properties of the asymptotic homology cycles of Sol Schwartzman. In particular, asymptotic homotopy is an invariant of topological conjugacy, and, in certain cases, of topological equivalence. We compute the asymptotic homotopy of those measure-preserving flows on Heisenberg manifolds which lift from the torus ${T^2}$ (Theorem 8.1), and then show how this invariant distinguishes up to topological equivalence certain of these flows which are indistinguishable homologically (Theorem 9.1). We also compute the asymptotic homotopy of those geodesic flows for Heisenberg manifolds which come from left invariant metrics on the Heisenberg group (Example 8.1), and then show how this invariant distinguishes up to topological conjugacy certain of these flows which are indistinguishable homologically.

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