On the equivariant Morse chain complex of the space of closed curves (Q1107183)

Closed geodesics on a compact Riemannian manifold M can be described as critical points of the energy functional on the free loop space \(\Lambda\) M. This space carries a canonical O(2)-action. We modify the concept of an equivariant (or G-) CW complex for a compact Lie group G in the case \(G=S^ 1\) resp. \(G=O(2)\). Using the energy functional as Morse function this allows the definition of the equivariant Morse chain complex of \(\Lambda\) M. As application we study homology classes in M and in the quotients \(\Lambda M/S^ 1\) and \(\Lambda\) M/O(2) for certain manifolds M. The paper is a condensed version of the author's thesis (``Der Äquivariante Morse-Kettenkomplex des Raums der geschlossenen Kurven'', Bonner Mathematische Schriften Nr. 178 (1987)).