Study Dolbeault cohomology on complex manifolds with torus action.
problem Describe Dolbeault cohomology of complex manifolds with torus action.
method Describe Dolbeault cohomology algebra of canonical foliation, provide dga model, prove Hodge decomposition.
result Hodge decomposition for basic Dolbeault cohomology proved.
Study torus orbifolds with two fixed points and their cohomology.
problem Understanding the topological and cohomological properties of torus orbifolds with two fixed points.
method Analyze the equivariant topological type and use results from [DKS] to compute integral equivariant cohomology.
result Results on generators and relations for the cohomology of torus orbifolds with two fixed points.
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
Study shows flat torus is identified by first cohomology group's dimension.
problem Identifying flat torus among RCD∗(0,N) spaces. method Analysis of the first cohomology group.
result Proves flat torus when first cohomology group dimension equals N.
Embeds cohomology of hyperkahler manifolds into torus cohomology.
problem Embedding cohomology of hyperkahler manifolds into torus cohomology.
method Kuga-Satake construction and embedding of graded cohomology spaces.
result Compatibility of embeddings with Hodge structures and Lie algebra actions.
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…
Study shows cohomological limits on extending actions on 3-manifold boundaries.
problem Limits to extending group actions on 3-manifold boundaries.
method Cohomological obstructions for C0-actions on 3-manifolds. result Cohomological obstructions prevent extending actions on certain 3-manifold boundaries.
Study non-negatively curved GKM orbifolds and their cohomology.
problem Characterize non-negatively curved GKM orbifolds and their cohomology.
method Analyze rational cohomology rings and isometric actions of finite groups.
result Rational cohomology rings of GKM orbifolds are isomorphic to model orbifolds.
We give definitions of cohomology determinants for compact, connected, orientable 3-manifolds. We also give formulae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology determinants are related to Turaev torsion, though the author hopes that they have othe…
Injective construction proves bounded cohomology dimensions.
problem Injectivity of Gambaudo--Ghys construction on bounded cohomology.
method Generalized Gambaudo--Ghys construction on bounded cohomology.
result Injectivity of the construction and infinite-dimensional bounded cohomology.
We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…
The n-dimensional torus is uniquely characterized by specific harmonic forms.
problem Characterizing the n-dimensional torus via harmonic forms.
method Analyzing closed 1-forms on the torus to determine unique properties.
result The n-dimensional torus is the unique manifold supporting a linearly independent set of (n-1) closed 1-forms whose product determines a non-zero cohomological class.
We describe the cohomology of a specific type of foliation on complex manifolds.
problem Computing the basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds.
method Using an Eilenberg-Moore spectral sequence and the formality of the Cartan model for the torus action.
result The basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold is similar to that of a complete simplicial toric variety.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …
Improved bounds on Z2-torus actions on positively curved manifolds.
problem Bounding the rank of Z2-tori for fixed point set components. method Lowered the rank bound and classified cohomology rings.
result Fixed point set components are classified by integral or Z2-cohomology rings. Discretizes Hodge-Dirac operators on a torus.
problem Capturing geometric aspects of continuum Hodge theory in discrete settings.
method Discrete exterior calculus framework, Hodge-Dirac and Laplace operators.
result Proves discrete Hodge decomposition theorem on combinatorial torus.
In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the prese…
Defines algebraic structures in Lagrangian Floer cohomology using differential forms.
problem Modeling algebraic structures in Lagrangian Floer cohomology.
method Defines two algebra structures using differential forms and a closed-open map, showing they coincide.
result Two algebra structures for the 2-dimensional Clifford torus coincide.
Classifies actions of tori on manifolds up to diffeomorphisms.
problem Classifying actions of tori on manifolds up to diffeomorphisms.
method Using triples (Q, λ, c) to classify actions, where Q is a manifold-with-corners, λ is a unimodular labelling, and c is a cohomology class.
result Classifies locally standard smooth actions of T up to equivariant diffeomorphisms.
This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S^2 union T^2, regarded as mo…
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions wi…
The paper disproves a generalized toral rank conjecture with various counter-examples.
problem The conjecture that the sum of Betti numbers of a compact manifold with a torus action is bounded by 2r. method Provided counter-examples of smooth nilpotent fibre bundles of nilmanifolds with torus fibres of rank r. result There are sequences of torus fibrations with total space cohomology dimensions converging to 0 as rank r increases. In this note we define a lifting of a local torus action modeled on the standard representation (we call it a local torus action for simplicity) to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with…
Let M be a symplectic manifold, equipped with a Hamiltonian action of a torus T. We give an explicit formula for the rational cohomology ring of the symplectic quotient M//T in terms of the cohomology ring of M and fixed point data. Under some restrictions, our formulas apply to integral cohomology. In certain …
Two de Rham complexes in diffeology are compared using a factor map.
problem Comparing two de Rham complexes in diffeology.
method Using a factor map to connect the two de Rham complexes and Čech--de Rham spectral sequence.
result Singular de Rham cohomology of irrational torus is isomorphic to tensor product of original de Rham cohomology and exterior algebra.
Complex Cayley Grassmannian is a singular variety with smooth singular locus.
problem Characterize the singularities of the Cayley Grassmannian.
method Defined a torus action and used it to prove singularity and cohomology properties.
result Singular locus is smooth and has the same cohomology as CP5. New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
Given a torus bundle Y over the circle and a cohomology class [ω]∈H2(Y;Z) which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of Y with twisted coefficients in the universal Novikov ring.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold M. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well.
Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …
We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness o…
We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …
The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.
problem Understanding the fixed points and cohomology of almost complex torus manifolds.
method Using directed labeled multigraphs and Hirzebruch genera to encode and analyze the manifolds.
result Almost complex torus manifolds have positive Todd genus and at least n+1 fixed points.
The paper proves conditions for 2-torus manifolds to be equivariantly formal.
problem Characterizing 2-torus manifolds as equivariantly formal.
method Proving 2-torus manifolds are equivariantly formal under specific conditions.
result 2-torus manifolds are equivariantly formal if and only if the action is locally standard and all faces of the orbit space are mod 2 acyclic.
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent tor…
moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originat…
The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.
problem Understanding K-contact manifolds with minimal closed Reeb orbits and their homeomorphism properties.
method Using Boothby-Wang fibration and Hamiltonian torus actions, the study constructs and analyzes K-contact manifolds.
result The existence of K-contact manifolds with minimal closed Reeb orbits that are not homeomorphic to spheres and have unique cohomology rings.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
problem Finding minimal triangulations of hyperbolic 3-manifolds.
method Characterization of low degree edges, layered solid torus subcomplexes, and 1-dimensional cohomology.
result Monodromy ideal triangulations of once-punctured torus bundles are minimal.
Abstract invariant cannot be expressed using various slice-torus invariants.
problem Cannot express Iida-Taniguchi's slice-torus invariant using other known invariants.
method Analysis of various known invariants and their properties.
result Iida-Taniguchi's slice-torus invariant cannot be realized as a linear combination of other invariants.
Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficien…
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.
problem Characterizing diffeomorphisms between nilmanifolds and smooth quasi-projective varieties.
method Analyzing cohomology and diffeomorphism properties of quasi-projective varieties and nilmanifolds.
result Nilmanifolds are diffeomorphic to trivial bundles over tori under certain conditions.
The paper proves non-triviality of certain classes in sphere bundle cohomology.
problem Proving non-triviality of powers of Euler and Pontryagin classes in sphere bundle cohomology.
method Using cobordism and free torus actions on manifolds, the paper constructs examples and proves non-triviality of classes.
result Powers of the Euler class and Pontryagin classes are non-trivial in the cohomology of the diffeomorphism group of odd-dimensional spheres.
The paper constructs symplectic forms on frame bundles and proves finite cohomologies.
problem Proving finiteness theorems for basic symplectic cohomologies.
method Construction of invariant 2-forms on the real symplectic group, symplectic form on quotient by a maximal torus, and lifting symplectic structure to frame bundles.
result Valid proof of finiteness theorems for basic symplectic cohomologies.
In this paper we define, for each aspherical orientable 3-manifold M endowed with a \emph{torus splitting} T\c, a 2-dimensional fundamental l1-class [M]T\c whose l1-norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). …