Related papers: Morse complexes and multiplicative structures

This paper explores the relation between the structure of fibre bundles akin to those associated to a closed almost nonnegatively sectionally curved manifold and rational homotopy theory.

This paper studies the homotopy-type of bi-filtrations of compact manifolds induced as the pre-image of filtrations of the plane for generic smooth functions f : M --> R^2. The primary goal of the paper is to allow for a simple description…

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and…

We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to…

Given a smooth function f on R^n and a submanifold M, we prove that the set of diagonal quadratic forms q such that the restriction of f+q to M is Morse is a dense set (in the n-dimensional space of diagonal quadratic forms). The standard…

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify…

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 by counting moduli spaces of perturbed Morse…

We study the relationship between the higher Massey products on the cohomology $H$ of a differential graded algebra, and the $A_\infty$ structures induced on $H$ via homotopy transfer techniques.

For a smooth manifold A, we consider the ordered configuration space F_k(AxR) of k distinct points in AxR. We obtain an explicit homotopy construction of the configuration space F_k(AxR) and of the (k-2)-fold suspension of F_k(A). Under…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1. For a subset X of M denote by D(M,X) the group of diffeomorphisms of M fixed on X. In this note we consider a special class F of smooth maps…

This paper introduces the notion of $n$-morphisms between two $A_\infty$-algebras, such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies between $A_\infty$-morphisms. The set…

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…

The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much…

We describe a weighted $A_\infty$-algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the…

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of…

We consider the Fukaya category associated to a basis of vanishing cycles in a Lefschetz fibration. We show that each element of the Floer cohomology of the monodromy around infinity gives rise to a natural transformation from the Serre…

We give a description of the structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$. These free boundary problems are models of phase transition and they are closely related to minimal hypersurfaces. We…

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to…

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the…

We construct A-infinity functors between Fukaya categories associated to monotone Lagrangian correspondences between compact symplectic manifolds. We then show that the composition of A-infinity functors for correspondences is homotopic to…