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Article Dans Une Revue Journal of Fixed Point Theory and Applications Année : 2023

Loop coproduct in Morse and Floer homology

Résumé

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. Firstly, 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. Secondly, 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. Thirdly, 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 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.

Dates et versions

hal-04058531 , version 1 (04-04-2023)

Identifiants

Citer

Kai Cieliebak, Nancy Hingston, Alexandru Oancea. Loop coproduct in Morse and Floer homology. Journal of Fixed Point Theory and Applications, 2023, 25 (2), pp.59. ⟨10.1007/s11784-023-01061-z⟩. ⟨hal-04058531⟩
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