Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive scalar curvature on a smooth manifold and its corresponding moduli spaces. In t…
We establish surgery formulas for filtration of the Heegaard Floer homology associated with p/q surgery on a null-homologous knot K in a three-manifold Y, induced by K_{p/q}. Here K_{p/q} is the core of the attached solid torus (which produces the surgery). This would generalize a result of Ozsvath and Szabo. We will r…
Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.
problem Finding homology 3-spheres with non-Morse-Bott Chern-Simons functions.
method Constructing specific surgeries on torus knots.
result Examples of homology 3-spheres with non-Morse-Bott Chern-Simons functions.
The paper calculates the Euler characteristic of regular spherical polygon spaces.
problem Determining the Euler characteristic of regular spherical polygon spaces.
method Constructing a manifold Xn and a function μ:XnoR such that μ−1(a)=Mn(a), determining the index of critical points, and using Morse surgeries. result Calculating the Euler characteristic χ(Mn(a)) for all a and odd n. We apply Lescop's construction of Z-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of "AL-paths" given in arXiv:1403.8030. We obtain an invariant Z^n of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over S1, whi…
This research connects topological changes to cosmic phenomena like black hole formation.
problem Understanding topological changes in cosmic phenomena.
method Using topological surgery and Morse functions to describe changes in 3-manifolds and their fundamental groups.
result New insights into natural phenomena through a topological perspective.
Solves an Arnold trivium problem using calculus and topology.
problem Finding critical points on a two-dimensional surface.
method Lagrange multipliers, Morse theory, Poincare-Hopf theorem.
result Determines the genus of a two-dimensional surface.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
Let K be a rationally null-homologous knot in a three-manifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgerie…
Study shows flexibility of homology groups of Reeb spaces of fold maps through surgery operations.
problem Understanding changes in homology groups of Reeb spaces of fold maps.
method Introduced surgery operations (bubbling operations) to fold maps and used elementary theory of sequences and continuous functions.
result Homology groups of Reeb spaces of fold maps constructed by iterations of these operations are flexible and can be represented as direct sums of original homology groups and finitely generated commutative groups.
Paper constructs fold maps with useful singular value sets.
problem Creating fold maps with specific singular value sets.
method Surgery operations to construct fold maps with crossings.
result Fold maps with singular value sets containing crossings.
It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in th…
The paper explores how topological methods can reveal insights into electric charge distributions on knots.
problem Understanding the qualitative behavior of electric potentials on knots.
method Geometric topology techniques applied to electrostatics.
result Proved a lower bound on the size of the critical set based on knot projections.
We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among…
In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differentia…
Study of fold maps and Reeb spaces via surgery operations.
problem Constructing explicit fold maps for complex manifolds.
method Surgery operations on fundamental fold maps to study Reeb spaces.
result Detailed study of cohomology rings of Reeb spaces for specific cases.
This paper restricts torsion subgroups in homology groups of Reeb spaces.
problem Understanding the structure of homology groups of Reeb spaces of fold maps.
method Explicitly studying the algebraic constraints on homology groups of Reeb spaces constructed by surgery operations.
result Explicit strong restrictions on the torsion subgroups of homology groups.
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator Dt on each regular level set Ct of a fixed Morse function defining this cobordism. We show that as we approac…
Defines Morse-Bott functions on manifolds with boundary and proves inequalities.
problem No specific problem stated; focuses on generalizing Morse theory.
method Defines Morse-Bott functions and proves inequalities for manifolds with boundary.
result Proves Morse-Bott inequalities for manifolds with boundary.
New method extends discrete Morse theory to simplicial complexes.
problem Discrete Morse theory on simplicial complexes.
method Morse shellings and compatible discrete Morse functions.
result Triangulated surfaces and manifolds have Morse shellable triangulations.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
Clarifies incorrect orientation assumptions in Morse-Bott inequalities.
problem Incorrect orientation assumptions in Morse-Bott inequalities.
method Provides explicit counterexamples and clarifies the origin of the mistake.
result Corrects the insufficient assumptions in Morse-Bott inequalities.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…
The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.
problem Analyzing cohomology of complex manifolds using Morse-Novikov theory.
method Establishing invariants, the Leray-Hirsch theorem, and blow-up formula for Dolbeault-Morse-Novikov cohomology.
result Established relations and stabilities of dimensions under complex structure deformations.
The paper introduces Morse theory for Lie groupoids and proves inequalities.
problem Defining Morse theory for Lie groupoids and studying their properties.
method Introducing Morse Lie groupoid morphisms and proving their Morita invariance.
result Established Morse theory for Lie groupoids and proved Morse inequalities.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. Generalizes Morse theory to n-categories using critical points and moduli spaces.
problem Constructing n-categories from Morse theory.
method Extending Cohen & Jones & Segal's flow category to n-categories by Morse theory on critical points and moduli spaces.
result The resulting structure is an 'almost strict' n-category.
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
Morse theory extended to noncompact manifolds with complex geometric data.
problem Extending Morse theory to noncompact manifolds with intricate geometric and homotopy data.
method Defining Morse homology for pairs of manifolds and related geometric/homotopy data, constructing a homotopy coherent diagram of linear maps, and showing it computes Morse homology.
result Morse homology can be computed using a chain complex derived from a homotopy coherent diagram.
Surveying Morse boundaries and stability results.
problem Understanding Morse boundaries and their stability.
method Review of existing literature.
result Compilation of known results on Morse boundaries and stability.
This paper equips Morse cochain complexes with A∞-algebra structures.
problem Equipping Morse cochain complexes with A∞-algebra structures. method Analogous to K. Fukaya's definition, this paper provides a detailed treatment of Abouzaid's approach.
result Provides a coherent and detailed treatment of Abouzaid's approach to Morse cochain complexes.
Classifies Morse boundaries of 3-manifold groups.
problem Classifying Morse boundaries of 3-manifold groups.
method Classifies Morse boundaries into 9 types based on geometric decompositions.
result 9 different homeomorphism types of Morse boundaries.
Exponential growth of stable subgroups in Morse geodesics.
problem Growth rates of stable subgroups in complex groups.
method Theory of automatic structures on Morse geodesics.
result Exponential growth of stable subgroups is faster than their infinite index stable subgroups.
The paper studies twisted Morse homology and cohomology on manifolds.
problem Computing homology and cohomology with local coefficients on manifolds.
method Morse theory, CW-complexes, de Rham cohomology, Lichnerowicz cohomology.
result Isomorphisms between different cohomology theories.
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
FPP preserves sublinear Morse boundaries in geodesic graphs.
problem Preserving sublinear Morse boundaries in FPP.
method First passage percolation on geodesic graphs with i.i.d. passage times.
result Sublinear Morse boundaries are invariant under FPP.
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.