Loop coproduct in Morse and Floer homology

TL;DR

This paper extends the isomorphism between symplectic homology of cotangent bundles and loop space homology, introducing new coproduct structures and a filtered chain isomorphism linking Floer and Morse complexes.

Contribution

It defines Rabinowitz loop homology via Morse theory, relates secondary coproducts under the Viterbo isomorphism, and introduces reduced loop homology with a new filtered isomorphism.

Findings

Rabinowitz loop homology product matches pair-of-pants product

Viterbo isomorphism intertwines coproducts on positive Floer and loop homology

Filtered chain isomorphism between Floer and Morse complexes using linear Hamiltonians

Abstract

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper we extend the scope of this isomorphism in several directions. First, we give a direct definition of {\em Rabinowitz loop homology} in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to {\em positive} Floer homology, resp.~loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce {\em reduced loop homology}, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. 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.