相关论文: Morse-Bott homology

In 1995, Cohen, Jones and Segal proposed a method of upgrading any given Floer homology to a stable homotopy-valued invariant. For a generic pseudo-gradient Morse-Bott flow on a closed smooth manifold $M$, we rigorously construct the…

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped…

We extend the Cohen-Jones-Segal construction of stable homotopy types associated to flow categories of Morse-Smale functions $f$ to the setting where $f$ is equivariant under a finite group action and is Morse but no longer Morse-Smale.…

Given two Morse functions $f, \mu$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, \eta)$ to $f(x) + \eta \mu(x)$. Take a product metric on $M \times…

We develop a theory of Morse homology and cohomology with coefficients in a derived local system, for manifolds and also more generally for colimits of spaces that have the homotopy type of manifolds, with a view towards Floer theory. The…

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical…

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…

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The…

Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or…

By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the…

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies, and the strict…

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,…

We study the rational homology of the Deligne--Mumford compactification $\overline{\mathcal M}_{g,n}$ of the moduli space of stable curves via a family of Morse functions, namely the $\text{sys}_T$ functions. Exploiting the geometric and…

We construct Hamiltonian Floer complexes associated to continuous, and even lower semi-continuous, time dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps…

We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting. In other words, we define framings for Morse-Bott analogues of flow categories and associate a stable homotopy type to this data. We use this to recover the stable…

We investigate the homology of cosheaves over finite simplicial complexes. After constructing the Mayer-Vietoris short exact sequence for this homology theory, we apply discrete Morse theory to this setting, defining the associated Morse…

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold satisfying non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of…

We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, epsilon. As an application, we show that an infinite family of topologically slice knots are independent in the smooth…

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about…

We prove a number of fundamental properties about instanton knot Floer homology. Our arguments rely on general properties of sutured Floer theories and apply also in the Heegaard Floer and monopole Floer settings, where many of our results…