arXiv research

The Morse complex is shown to be an infinite functor.

The article details Fukaya's A∞-structure of Morse complexes.

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse funct…

Survey on SYZ mirror symmetry using Morse theory.

New $A_\infty$ structures derived from equivariant de Rham complex for $S^1$-action.

Research resolves sign conventions in Floer theory for Morse-Bott case.

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct $A_\infty$-category structure on the relative Morse complex…

Proves immediate transversality for conic singularities.

In this paper, we extend the Witten-Helffer-Sjöstrand theory from Morse functions to generalized Morse functions. In this case, the spectrum of the Witten deformed Laplacian $Δ(t)$, for large t, can be seperated into the small eigenvalues (which tend to 0 as $t\rightarrow\infty$), large and very large eigenvalues (both…

Study Morse theory on loop spaces and Hecke algebras.

New pairing defined from Morse complexes for compact manifolds.

We prove a generalized version of the classic deformation lemma from Morse Theory that considers functions going to $-\infty$ at a compact set, and allowing the lower value of the deformation to be $-\infty$. The result is valid for a class of functions satisfying a suitable growth condition.

The paper studies connectivity properties of Morse complexes as simplicial complexes grow.

Fix a tangential structure $θ: B \longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\mathbf{Cob}^{\text{mf}, k}_θ$, where morphisms are given by $θ$-cobordisms $W: P \rightsquigarrow Q$ equipped with a choice of proper Morse function $h_{W}: W \longr…

Study of critical points in Ginzburg-Landau approximation with stability results.

A new approach to Morse theory using folded ribbon trees.

We study algebraic structures ($L_\infty$ and $A_\infty$-algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories. We show that such structures can be systematically produced in any number of dimensions by using the geometry of seconda…

We consider systems $(M,ω,g)$ with $M$ a closed smooth manifold, $ω$ a real valued closed one form and $g$ a Riemannian metric, so that $(ω,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $ρ(ω,g)\in[0,\infty]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a …

Develops Morse theory for commuting gradient-like vector fields.

In this paper, the following three are shown. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$. (2) For a stable convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is stable. (3) Let $γ: S…

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

We present an elementary and self-contained construction of $A_\infty$-algebras, $A_\infty$-bimodules and their Hochschild homology and cohomology groups. In addition, we discuss the cup product in Hochschild cohomology and the spectral sequence of the length filtration of a Hochschild chain complex. $A_\infty$-structu…

Generalizes Giroux's result to higher dimensions and applies to contact manifolds.

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex in…

The paper studies the automorphisms of Kronrod-Reeb graphs for Morse functions on a 2-sphere.

Study knot Floer cohomology using hyperbolic metrics and wrapped Fukaya categories.

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookright…

The paper studies heat kernel asymptotics and proves Morse inequalities.

The paper proves Morse estimates for translated points on unit tangent bundles.

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $χ(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\c…

Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as op…

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $χ(M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a …

Novikov homology refined for 2D integer group coverings.

The paper examines how the topology of level sets changes with critical points in Morse theory.

For all $n$, we define the $n$-dimensional critical catenoid $M_n$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $\Bbb{R}^{n+1}$. We show that the Morse index $\text{MI}(n)$ of $M_n$ satisfies the following asymptotic estimate as …

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1…

Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.

The paper proves bounds on the Morse index of free boundary minimal hypersurfaces.

One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces $J^1M$, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product $μ_2$. To define the product, moduli…

We compute the higher $Σ$-invariants $Σ^m(F_{n,\infty})$ of the generalized Thompson groups $F_{n,\infty}$, for all $m,n\ge 2$. This extends the $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ case done by Kochloukova. Our approach differs from those used in the $n=2$ and $m=2$ cases; we look at the …

The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.

The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M …

Complete description of BNSR invariants for Lodha-Moore groups, proving finiteness properties.

New category theory for complex projective plane sections.

Exponential growth of stable subgroups in Morse geodesics.

Study Morse functions on projective plane using Reeb graphs.

New Q-Newton's method avoids saddle points and converges quadratically.