Lie monoid actions in Morse theory

Establish that for compact Lie monoids G, G′, H, H′ acting on closed smooth manifolds X and X′ with morphisms of Lie monoids φ: G→G′ and ψ: H→H′ and a (φ, ψ)-bi-equivariant map f: X→X′, the following hold at the chain level: (i) the Morse complexes CM(G), CM(G′), CM(H), and CM(H′) admit forest bialgebra (f-bialgebra) structures; (ii) the Morse complexes CM(X) and CM(X′) admit u-bimodule structures over the corresponding f-bialgebras; (iii) φ and ψ induce morphisms of f-bialgebras on Morse chains; and (iv) f induces a (φ*, ψ*)-bi-equivariant morphism of u-bimodules CM(X)→CM(X′).

Background

The paper introduces forest biassociahedra and forest bimultiplihedra to encode chain-level operations arising from interactions between Lie monoid actions and the multiplicative structures in Morse theory. These moduli spaces parametrize operations indexed by multi-indices and lead to the notion of forest bialgebras (f-bialgebras) and their u-bimodules.

The conjecture asserts that when a Lie monoid acts on a manifold, the Morse complexes of the monoids themselves carry f-bialgebra structures, the Morse complex of the manifold carries a compatible u-bimodule structure, and bi-equivariant maps induce morphisms respecting these structures. This provides a chain-level enhancement of the classical homological bialgebraic picture (Pontryagin product and coproduct) for Lie monoids.