Classifying Spaces of Moduli Spaces of Morse Smale Flows

Classifying spaces of groups were introduced by Milnor in his study of fibrations. Segal has generalized Milnor's construction to define the classifying spaces of topological categories. Motivated by the Floer Theory (in which one studies the moduli spaces of pseudo-holomorphic strips) Cohen, Jones, and Segal have studied a topological category of a Morse Smale flow on compact manifold whose objects are stationary points of the flow and whose morphism are moduli spaces of flow-lines. They proved that the classifying space of the topological category of a Morse Smale flow is homeomorphic to the underlying manifold. In this work we study classifying spaces and moduli spaces of Morse Smale flows from dynamical systems point of view. In particular, we improve the Cohen, Jones, and Segal Theorem by showing that the Morse Smale flow on the manifold is topologically conjugate to a natural projective flow on the classifying space. This work will be of interest to everyone studying or doing research in dynamical systems, geometry, and topology or related areas.