Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory I

Abstract

Elaborating on works by Abouzaid and Mescher, we prove that for a Morse function on a smooth compact manifold, its Morse cochain complex can be endowed with an ΩBAs-algebra structure by counting moduli spaces of perturbed Morse gradient trees. This rich structure descends to its already known A∞-algebra structure. We then introduce the notion of ΩBAs-morphism between two ΩBAs-algebras and prove that given two Morse functions, one can construct an ΩBAs-morphism between their associated ΩBAs-algebras by counting moduli spaces of two-colored perturbed Morse gradient trees. This morphism induces a standard A∞-morphism between the induced A∞-algebras. We work with integer coefficients, and provide to this extent a detailed account on the sign conventions for A∞ (resp. ΩBAs)-algebras and A∞ (resp. ΩBAs)-morphisms, using polytopes (resp. moduli spaces) which explicitly realize the dg-operadic objects encoding them. Our proofs also involve building at the level of polytopes an explicit functor from the category of ΩBAs-algebras to the category of A∞-algebras, drawing from a result by Markl and Shnider. This paper is adressed to people acquainted with either symplectic topology or algebraic operads, and written in a way to be hopefully understood by both communities. It comes in particular with a detailed survey on operads, A∞-algebras and A∞-morphisms, the associahedra and the multiplihedra, as well as some details on the usual techniques used in symplectic topology to define algebraic structures on geometrical (co)chain complexes. It moreover lays the basis for a second article in which we solve the problem of finding a satisfactory homotopic notion of higher morphisms between A∞-algebras and between ΩBAs-algebras, and show how this higher algebra of A∞ and ΩBAs-algebras naturally arises in the context of Morse theory.The associahedron K 4 and the multiplihedron J 3 ...

The diagonal of the operahedra

“…Future directions. The family of generalized permutahedra include an example of fundamental importance in symplectic topology: the multiplihedra [Maz21a,Maz21b]. A cellular approximation of the diagonal of the multiplihedra would allow one to define the tensor product of 𝐴 ∞ -categories [LOT20].…”

Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory II

“…These moduli spaces come in fact with thinner cell decompositions, called their ΩBAs-cell decompositions : the cell decomposition by stable ribbon tree type for T m , and the cell decomposition by stable gauged ribbon tree type for CT m . These thinner decompositions provide another operadic model for strong homotopy associative algebras with morphisms preserving the product up to homotopy between them : the standard operad ΩBAs and the operadic bimodule ΩBAs − Morph introduced in [Maz21]. We show moreover in [Maz21] that one can naturally shift from the ΩBAs to the A ∞ framework via a morphism of operads A ∞ → ΩBAs and a morphism of operadic bimodules…”

“…These thinner decompositions provide another operadic model for strong homotopy associative algebras with morphisms preserving the product up to homotopy between them : the standard operad ΩBAs and the operadic bimodule ΩBAs − Morph introduced in [Maz21]. We show moreover in [Maz21] that one can naturally shift from the ΩBAs to the A ∞ framework via a morphism of operads A ∞ → ΩBAs and a morphism of operadic bimodules…”

“…Summary and results of article I . This article is the direct sequel to [Maz21]. We thus begin by summarizing our first article, after which we outline the main results and constructions carried out in the present paper.…”

“…, x m ). We prove in [Maz21] that under generic assumptions on the choice of perturbation data, these moduli spaces are in fact orientable manifolds of finite dimension. If they have dimension 1, they can moreover be compactified to 1-dimensional manifolds with boundary, whose boundary is modeled on the top dimensional strata in the boudary of the compactified moduli space T m .…”