Mathematics – Geometric Topology
Scientific paper
2007-02-14
Mathematics
Geometric Topology
105 pages, 20 figures. Corrected typos and added a new section on the classification of gradient like tame flows
Scientific paper
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $\Phi: \mathbb{R}\times X\to X$ 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 the Morse-Smale condition is equivalent to 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 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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