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.
Paper constructs Thom-Smale complex using instantons from Morse functions.
problem Constructing Thom-Smale complex for Morse functions.
method Analytic instanton construction using eigenspaces of mapping cone Laplacian.
result Instanton complex is cochain isomorphic to Thom-Smale complex.
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)∈[0,∞] and improve Morse-Novikov theory by showing that the Novikov complex comes from a …
New construction of Fukaya-Seidel categories using complex gradient flow equation.
problem Constructing Fukaya-Seidel categories for specific models.
method Using the complex gradient flow equation and neck-stretching limits.
result Alternative proof of Seidel's spectral sequence for Lagrangian Floer cohomology.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Geometrically interprets a duality theorem linking cochain and chain complexes.
problem Understanding a complex duality theorem in geometric terms.
method Introduces a chain isomorphism involving simplicial and cellular complexes.
result Establishes a geometric interpretation of Ranicki duality.
New complexes derived from any filtered cochain complex compute the same cohomology.
problem Constructing cohomologically equivalent subcomplexes from filtered cochain complexes.
method Presenting a general construction that produces subcomplexes from any filtered cochain complex of finite depth.
result The construction of subcomplexes depends only on the filtration up to isomorphism.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We a…
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. We compute Steenrod squares on Khovanov homology.
problem Computing Steenrod squares on Khovanov homology.
method Stable cup-i products on cochain complexes of augmented semi-simplicial objects in the Burnside category.
result Explicit formulas for cohomology operations on Khovanov homology.
The paper computes KV cochain differentials and their geometric implications.
problem Deformation theory of flat and torsion-free affine connections.
method Explicit computation of KV cochain differentials and their relations to geometric transformations.
result KV algebra with non-vanishing second cohomology group.
Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra sl2m. We exhibit bijections between a set of generators for the Sei…
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
Paper explores connection cochain in abelian extensions and its relation to connection forms.
problem Understanding the connection cochain in abelian extensions.
method Apply Moriyoshi's connection cochain concept to abelian extensions and relate it to connection 1-forms.
result Established the relationship between connection cochain and connection 1-forms in abelian extensions.
Study vector bundles over Lie groupoids, controlling their deformations.
problem Understanding deformations of vector bundles over Lie groupoids.
method Attach cochain complexes to VB-groupoids to control deformations, discuss Morita invariance and van Est theorem.
result Fundamental features of VB-groupoids' deformations, including Morita invariance and van Est theorem.
Develops combinatorial theory of vector bundles on simplicial complexes.
problem Creating a discrete theory for vector bundles and connections on simplicial complexes.
method Introduces discrete exterior covariant derivative and applies it to various geometric objects.
result Flat discrete connections yield a cochain complex computing twisted de Rham cohomology.
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …
New models for B-type topological theories using complex functions.
problem Constructing open-closed topological field theories for non-compact Calabi-Yau manifolds.
method Differential models using cochain level data, including Dolbeault algebras and categories.
result Most axioms satisfied on cohomology, conjecture remaining axioms hold.
Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We pr…
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.
In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner p…
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.
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
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.
Introduces θ-almost twisted Poisson structures and their cohomology.
problem Characterizing and understanding θ-almost twisted Poisson structures. method Definition and construction of θ-almost twisted Poisson structures, Lie-Rinehart algebra, cochain complex, and cohomology. result Definition and construction of θ-almost twisted Poisson cohomology. We characterize Lie group actions for which there exists, at least locally, an evaluation map that defines a cochain map from the differential complex of invariant forms on a manifold to the De Rham complex for the quotient.
The Van Est homomorphism for a Lie groupoid G⇉M, as introduced by Weinstein-Xu, is a cochain map from the complex C∞(BG) of groupoid cochains to the Chevalley-Eilenberg complex C(A) of the Lie algebroid A of G. It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from…
We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,ω) with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)…
A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in Rn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m in Rn. The main purpose of the present paper is to generalize Wolfe's theorem to the se…
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.
New pairing defined from Morse complexes for compact manifolds.
problem Defining a pairing for compact manifolds with Morse functions.
method Constructing Morse complexes and a short exact sequence.
result Induces the intersection product in homology.
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.
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular N-cube chains when the function is constant. We show that the ho…
Paper addresses hidden faces in configuration space integrals for embeddings.
problem Understanding hidden faces in configuration space integrals for long embeddings.
method Modified configuration space integrals incorporating acyclic bar complex of a dg algebra.
result Cochain map from new graph complex to de Rham complex of embeddings modulo immersions.
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…
The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.
Geometric cohomology model uses co-oriented maps to define a product structure.
problem Constructing a geometric model for cohomology of smooth manifolds.
method Develops a cochain complex model based on co-oriented smooth maps, focusing on their pull-back product structure.
result Geometric cochains with a partially defined product structure induce the cup product in cohomology.
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
The Morse complex is shown to be an infinite functor.
problem Understanding the structure of Morse complexes as infinite functors.
method Showed the Morse complex of a compact Lie monoid can be given the structure of an f-bialgebra and defined an ∞-functor.
result Obtained two other ∞-functors mapping manifolds and actions to their Morse complexes.
The paper studies connectivity properties of Morse complexes as simplicial complexes grow.
problem Understanding connectivity of Morse complexes as simplicial complexes evolve.
method Bestvina-Brady Morse theory applied to a generalized Morse complex.
result Proves M(Δ) becomes arbitrarily highly connected as Δ grows. The paper connects group extensions, cochains, and spectral sequences.
problem Understanding the relationship between group extensions and spectral sequences.
method Using connection cochains, the paper derives a formula for the extension class.
result A formula clarifies the relation among connection cochains, extension classes, and the LHS spectral sequence.
Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2…
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.
We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the "straightening" of the coarse laminations defined in [Ca], by using minimal surface techniques. Moreover, whe…
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
A Morse complex for Axiom A flows on smooth manifolds.
problem Constructing a finite-dimensional cohomological complex for Axiom A flows.
method Defining anisotropic Sobolev spaces and spectral projectors.
result The cohomology of the constructed complex is isomorphic to De Rham cohomology.
This paper categorifies Morse theory for manifolds with boundaries.
problem Categorifying Morse theory for manifolds with boundaries.
method Defining a relative Morse complex using handlebody decomposition and constructing an A∞-category structure. result The homology of the relative Morse complex is isomorphic to the relative singular homology.