Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic homotopy cycles for flows and $ \Pi \sb 1$ de Rham theory


Authors: Diego Benardete and John Mitchell
Journal: Trans. Amer. Math. Soc. 338 (1993), 495-535
MSC: Primary 58F17; Secondary 22E25, 57R99, 58A12, 58F11, 58F25
MathSciNet review: 1093216
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F17, 22E25, 57R99, 58A12, 58F11, 58F25

Retrieve articles in all journals with MSC: 58F17, 22E25, 57R99, 58A12, 58F11, 58F25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1093216-6
PII: S 0002-9947(1993)1093216-6
Keywords: Nonabelian de Rham theory, flows, asymptotic homology, asymptotic homotopy, topological conjugacy, topological equivalence, Heisenberg manifolds, geodesic flows, product integral, nilpotent Lie groups
Article copyright: © Copyright 1993 American Mathematical Society