2021/02/12 by Mazuir, Thibaut
#18M70 #18N70 #37D15 #52B05 #52B11 #53D30 #Algebraic Topology (math.AT) #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2102.06654
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 ΩB As-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 ΩB As-morphism between two ΩB As-algebras and prove that given two Morse functions, one can construct an ΩB As-morphism between their associated ΩB As-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. ΩB As)-algebras and A_∞ (resp. ΩB As)-morphisms, using polytopes (resp. moduli spaces) which explicitly realize the dg-operadic objects encoding them. Our proofs also involve showing at the level of polytopes that an ΩB As-morphism between ΩB As-algebras naturally induces an A_∞-morphism between A_∞-algebras. This paper comes in particular with a short survey on operads, A_∞-algebras and A_∞-morphisms, the associahedra and the multiplihedra. All the details on transversality, gluing maps, signs and orientations for the moduli spaces defining the algebraic structures on the Morse cochains are thorougly carried out. 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 ΩB As-algebras, and show how this higher algebra of A_∞ and ΩB As-algebras naturally arises in the context of Morse theory.