Multiplicative structures on cones and duality

TL;DR

This paper explores multiplicative algebraic structures on cones in Floer theory, providing new descriptions of Rabinowitz Floer homology and a novel proof of Poincaré duality.

Contribution

It introduces two new algebraic frameworks for Floer chains and their duals, enhancing understanding of multiplicative structures and duality in Floer homology.

Findings

New algebraic structures for Floer chains and duals

A novel proof of Poincaré duality in Floer homology

Enhanced description of Rabinowitz Floer homology multiplicative structure

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 Poincar\'e 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 $\mathcal{A}$.