Related papers: C^0-topology in Morse theory

Michael Farber introduced the Lusternik-Schnirelmann category cat$(M,\xi)$ for the pair of finite CW complex $M$ and first-order cohomology $\xi$. It is inspired by the Morse-Novikov theory, which is a closed 1-form version of the Morse…

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of…

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain…

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick…

In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the…

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer…

We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-$0$ curves on Hamiltonian Floer theory; this operad is equivalent to the framed $E_2$ operad. We formulate the construction in the…

We prove a compactness result for gradient flow lines in a general set-up which comprises both the situation of Morse gradient flow lines as well as Floer cylinders converging to a critical submanifold respectively. For the compactness…

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation…

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that,…

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy…

Let N be a closed four dimensional manifold which admits a self-indexing Morse function f with only 3 critical values 0,2,4, and a unique maximum and minimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. We construct…

The classical Morse--Sard theorem claims that for a mapping $v:\mathbb R^n\to\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\ge n-m$. Here the critical set, or $m$-critical set is…

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely…

This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out…

Let $f:M \to \mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds…

We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$. A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on the order…

We study smooth {\sf traversing} vector fields $v$ on compact manifolds $X$ with boundary. A traversing $v$ admits a Lyapunov function $f: X \to \Bbb R$ such that $df(v) > 0$. We show that the trajectory spaces $\mathcal T(v)$ of {\sf…

Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…