Bialgebras, and Lie monoid actions in Morse and Floer theory, I

TL;DR

This paper introduces new geometric structures called forest biassociahedra and bimultiplihedra, which serve as parameter spaces for operations in Morse and Floer theory related to Lie group actions.

Contribution

It defines novel moduli spaces generalizing biassociahedra, studies their boundary structure, and proposes algebraic frameworks for Lie group actions on Morse and Floer chains.

Findings

Introduction of forest biassociahedra and bimultiplihedra as moduli spaces.

Development of algebraic notions of f-bialgebras and bimodules.

Formulation of conjectures on Lie group actions in Morse and Floer theory.

Abstract

We introduce a new family of oriented manifolds with boundaries called the forest biassociahedra and forest bimultiplihedra, generalizing the standard biassociahedra. They are defined as moduli spaces of ascending-descending biforests and are expected to act as parameter spaces for operations defined on Morse and Floer chains in the context of compact Lie group actions. We study the structure of their boundary, and derive some algebraic notions of ``$f$-bialgebras'', as well as related notions of bimodules, morphisms and categories. This allows us to state some conjectures describing compact Lie group actions on Morse and Floer chains, and on Fukaya categories.