Multiplicative structures on cones and duality

Abstract

We initiate the study of multiplicative structures on cones and show that cones of Floer continuation maps fit naturally in this framework. We apply this to give a new description of the multiplicative structure on Rabinowitz Floer homology and cohomology, and to give a new proof of the Poincaré duality theorem which relates the two. The underlying algebraic structure admits two incarnations, both new, which we study and compare: on the one hand the structure of $A_2^{+}$-algebra on the space $\mathcal{A}$ of Floer chains, and on the other hand the structure of $A_2$-algebra involving $\mathcal{A}$, its dual ${\mathcal{A}}^{\vee }$ and a continuation map from ${\mathcal{A}}^{\vee }$ to $\mathcal{A}$.

90 pages, 43 figures. Compared to the previous version, the main changes concern the introduction, which has been significantly expanded. This version to appear in the Journal of Symplectic Geometry

References in corpus (2)

• Loop coproduct in Morse and Floer homology
• Poincaré duality for loop spaces

Cited by in corpus (2)

• Poincaré duality for loop spaces
• Loop coproduct in Morse and Floer homology