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Tame flows

About this Title

Liviu I. Nicolaescu, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618.

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 208, Number 980
ISBNs: 978-0-8218-4870-8 (print); 978-1-4704-0594-6 (online)
Published electronically: July 13, 2010
MathSciNet review: 2683658
Keywords:Definable spaces and maps, real analytic manifolds and maps, Grassmannians, Morse flows, Morse-Smale condition, stratifications, Whitney and Verdier conditions, Conley index, simplicial spaces, topology of posets, currents, finite volume flows
MSC (2000): Primary 37B30; Secondary 37D15, 55P05, 55U10, 57Q05, 57R05, 58A35

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Table of Contents


  • Introduction
  • Chapter 1. Tame spaces
  • Chapter 2. Basic properties and examples of tame flows
  • Chapter 3. Some global properties of tame flows
  • Chapter 4. Tame Morse flows
  • Chapter 5. Tame Morse-Smale flows
  • Chapter 6. The gap between two vector subspaces
  • Chapter 7. The Whitney and Verdier regularity conditions
  • Chapter 8. Smale transversality and Whitney regularity
  • Chapter 9. The Conley index
  • Chapter 10. Flips/flops and gradient like tame flows
  • Chapter 11. Simplicial flows and combinatorial Morse theory
  • Chapter 12. Tame currents
  • Appendix A. An “elementary” proof of the generalized Stokes formula
  • Appendix B. On the topology of tame sets


The tame flows are “nice” flows on “nice” spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi : \mathbb {R}\times X\rightarrow X$ on pfaffian set $X$ is tame if the graph of $\Phi $ is a pfaffian subset of $\mathbb {R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove that the flow determined by the gradient of a generic real analytic function with respect to a generic real analytic metric is tame. The typical tame gradient flow satisfies the Morse-Smale condition, and we prove that in the tame context, under certain spectral constraints, the Morse-Smale condition implies the fact that the stratification by unstable manifolds is Verdier and Whitney regular. We explain how to compute the Conley indices of isolated stationary points of tame flows in terms of their unstable varieties, and then give a complete classification of gradient like tame flows with finitely many stationary points. We use this technology to produce a Morse theory on posets generalizing R. Forman’s discrete Morse theory. Finally, we use the Harvey-Lawson finite volume flow technique to produce a homotopy between the DeRham complex of a smooth manifold and the simplicial chain complex associated to a triangulation.

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