Article
MSC:
18M70,
18N70,
37D15,
52B05,
52B11,
53D30 | MR 4918786 | Zbl 08141784 | DOI: 10.21136/HS.2025.03
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Keywords:
Morse theory; operads; homotopy theory; polytopes; symplectic topology; combinatorics
Morse theory; operads; homotopy theory; polytopes; symplectic topology; combinatorics
Summary:
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 $\Omega B As$-algebra structure through counts of perturbed Morse gradient trees. This rich structure descends to its already known $A_\infty$-algebra structure. We then introduce the notion of $\Omega B As$-morphism between two $\Omega B As$-algebras and prove that given two Morse functions, one can construct an $\Omega B As$-morphism between their associated $\Omega B As$-algebras through counts of 2-colored perturbed Morse gradient trees. This continuation morphism is a quasi-isomorphism and induces a standard $A_\infty$-morphism between the induced $A_\infty$-algebras. We work with integer coefficients, and provide to this extent a detailed account on the sign conventions for $A_\infty$-algebras, $\Omega B As$-algebras, $A_\infty$-morphisms and $\Omega B As$-morphisms, using polytopes and moduli spaces of metric trees which explicitly realize the dg operadic objects encoding them. Our proofs also involve showing at the level of polytopes that an ΩBAs-morphism between $\Omega B As$-algebras naturally induces an $A_\infty$-morphism between $A_\infty$-algebras. This paper is adressed to people acquainted with either differential topology or algebraic operads, and written in a way to be hopefully understood by both communities. It comes in particular with a short survey on operads, $A_\infty$-algebras and $A_\infty$-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 notion of higher morphisms between $A_\infty$-algebras and between $\Omega B As$-algebras, and show how this higher algebra of $A_\infty$ and $\Omega B As$-algebras provides a natural framework to give a higher categorical meaning to the fact that continuation morphisms in Morse theory are well-defined up to homotopy at the chain level.
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