The space of paths in complex projective space with real boundary conditions

We compute the integral homology of the space of paths in $\mathbb{C}P^n$ with endpoints in $\mathbb{R}P^n$, $n \ge 1$ and its algebra structure with respect to the Pontryagin-Chas-Sullivan product with $\mathbb{Z}/2$-coefficients.Comment: 33 pages, 6 figure

Poincar\'e duality for loop spaces

We prove a Poincar\'e duality theorem with products between Rabinowitz Floer homology and cohomology, for both closed and open strings. This lifts to a duality theorem between open-closed TQFTs. Specializing to the case of cotangent bundles, we define extended loop homology and cohomology and explain from a unified perspective pairs of dual results which have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, level-potency, and homotopy invariance. We extend the loop cohomology product to include constant loops. We prove a relation conjectured by Sullivan between the loop product and the extended loop homology coproduct as a consequence of associativity for the product on extended loop homology.Comment: 87 pages, 14 figure

Loop coproduct in Morse and Floer homology

By a well-known theorem first proved by Viterbo, the Floer homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. We prove that, when restricted to positive Floer homology resp. loop space homology relative to the constant loops, this isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. The proof uses compactified moduli spaces of punctured annuli. We extend this result to reduced Floer resp. loop homology (essentially homology relative to a point), and we show that on reduced loop homology the loop product and coproduct satisfy Sullivan's relation. Along the way, we show that the Abbondandolo-Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism by using linear Hamiltonians and the square root of the energy functional.Comment: 76 pages, 17 figure