17 research outputs found
The space of paths in complex projective space with real boundary conditions
We compute the integral homology of the space of paths in
with endpoints in , and its algebra structure with
respect to the Pontryagin-Chas-Sullivan product with
-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