Defines log Floer cohomology for symplectic surfaces with a degenerate part.
problem Extending Floer cohomology to degenerate symplectic structures.
method Definition of log Floer cohomology for oriented log symplectic surfaces.
result Log Floer cohomology is invariant under isotopies and isomorphic to log de Rham cohomology for a single Lagrangian.
We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.
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.
Study cohomology of surfaces with punctures and boundaries, proving bounds on rational cohomology.
problem Understanding cohomology of surfaces with punctures and boundaries.
method Two proofs showing congruence subgroups have enormous rational cohomology.
result Bounds on cohomology are super-exponential in number of punctures and boundary components.
Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
Injective map from top cohomology of moduli spaces to handlebodies.
problem Understanding cohomology of moduli spaces of surfaces and handlebodies.
method Constructing a classifying space for handlebody mapping class group.
result Top weight cohomology of moduli spaces maps injectively into handlebodies.
Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.
problem Understanding the cohomology structure of SU(2) character varieties of surfaces.
method Explicit description of Poincaré duals of cohomology generators.
result An explicit description of the Poincaré dual of each generator of the rational cohomology ring.
Study on cohomological dimension of surface terms, answering Farb's question.
problem Determining the cohomological dimension of specific surface terms.
method Analyzing the Johnson filtration of closed, orientable surfaces of genus g≥2. result The kth term of the Johnson filtration has cohomological dimension 2g−3 for all k≥3 and g≥2. In this note, we compute the Poisson cohomology groups for any Poisson Del Pezzo surface.
Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.
problem Determine the p-primary component of Farrell cohomology for non-orientable surfaces. method Classify subgroups of order p using topological equivalence adapted to surfaces with marked points. result Determine the p-primary component of Farrell cohomology for non-orientable surfaces. We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev's method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of P2. A consequence of our theorem is that over the finite field Fq, away from finitely…
We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface Σ and the Donaldson invariants of the algebraic surface Σ×P1. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. …
We determine the quantum cohomology of the moduli space of odd degree rank two stable vector bundles over a Riemann surface Σ of any genus. This work together with dg-ga/9710029 prove that this quantum cohomology is isomorphic to the instanton Floer cohomology of the three manifold Σ×S1. (Note: There is some…
Study calculates integral cohomology of non-orientable infinite type surfaces.
problem Computing the first integral cohomology group of non-orientable infinite type surfaces.
method Alexander method, isomorphism to automorphism group, topological rigidity of curve graph, semi-direct product structure.
result First integral cohomology group computed for non-orientable infinite type surfaces.
New quandles classify surfaces, with infinite cohomology.
problem Classifying closed orientable surfaces using quandles.
method Introduced new quandle families, analyzed their cohomology, and computed their idempotents.
result Second bounded quandle cohomology is infinite-dimensional.
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
The study examines stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.
problem Stability of fibres on Hopf surfaces as harmonic maps and minimal surfaces.
method Construction of Hermitian metrics on Hopf surface and analysis of fibres as harmonic maps and minimal surfaces.
result Two toric fibres are stable minimal surfaces, while others are unstable.
Study Euler class of surface bundles with nontrivial results.
problem Euler class of infinite-type surface bundles with a section.
method Analyzing smooth orientable infinite-type surface bundles with a section.
result Nontrivial Euler class and dependence on genus and end types.
New method distinguishes knots and knotted surfaces.
problem Distinguishing knots and knotted surfaces.
method Twisted set-theoretic Yang-Baxter solutions and Alexander numbering.
result Distinguished 2-twist spun trefoil from its reverse. Study on dimensions of mapping class groups of non-orientable surfaces.
problem Determining dimensions of mapping class groups for non-orientable surfaces.
method Analyzing cohomological, geometric, and virtual dimensions.
result Equal dimensions for Ng when geq4,5. Surface bundles' signatures tied to their Euler characteristics.
problem Relating signatures and Euler characteristics of surface bundles.
method Using bounded cohomology and Morita's results.
result The inequality ∣3σ(E)∣≤∣χ(E)∣ for surface bundles. We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to Z factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.
Compute group cohomology for mapping class group with non-symplectic coefficients.
problem Compute group cohomology for mapping class group with non-symplectic coefficients.
method Compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group.
result First group cohomology computation for the mapping class group with non-symplectic coefficients.
We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
problem Characterizing extensions of augmented racks and constructing invariants for surfaces.
method Characterization of rack extensions through fibrant and additive cohomology, construction of invariants using cocycles.
result Characterization of extensions of augmented racks and construction of surface invariants.
Simply connected moduli space of Ricci flat metrics on K3 surfaces.
problem Understanding the topology of Ricci flat metrics on K3 surfaces.
method Analyzing the moduli space of metrics with unit volume.
result The moduli space is simply connected and has cohomology matching the automorphism group.
Researchers compute Dolbeault cohomology of Endo-Pajitnov manifolds.
problem Computing Dolbeault cohomology for a new class of non-Kähler manifolds.
method Computed Dolbeault cohomology using the Hodge decomposition.
result Endo-Pajitnov manifolds satisfy the Hodge decomposition at the level of dimensions.
Study shows periodic cohomology of non-orientable surface mapping class groups for odd primes.
problem Investigating periodic cohomology of non-orientable surface mapping class groups for odd primes.
method Using Yagita invariant, cohomology classes, Nielsen realization theorem, and properties of cyclic subgroups of order p.
result The p-period of Ngk is bounded below by 4 when Ngk has p-periodic cohomology, g⩾3 and k⩾0. Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
Thurston's spine dimension exceeds virtual cohomological dimension.
problem Understanding the dimension of Thurston's spine in moduli space.
method Analyzing closed hyperbolic surfaces and their systoles.
result The dimension of the set of surfaces is at least \(5-\varepsilon\) for large genus.
The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign…
Study how singular fibrations affect Poisson cohomology in 4D.
problem Understanding Poisson cohomology in singular fibrations.
method Analyzing the monodromy effect on Poisson cohomology groups.
result Describe the effect of monodromy on Poisson cohomology groups.
Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.
problem Defining and analyzing analytic lattice cohomology for isolated singularities.
method Using a good resolution of the singularity, proving independence of resolution choice, and relating to Hodge spectral numbers.
result Independence of analytic lattice cohomology from the choice of resolution and connection to Hodge spectral numbers.
We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces S0, S+, S− and classify the locally conformally Kähler (and the tamed loc…
This paper studies symplectic structures on elliptic surfaces with positive Euler number.
problem Determining symplectic representatives for cohomology classes on elliptic surfaces.
method Analyzes the symplectic cone for elliptic surfaces with positive Euler number.
result Characterizes the symplectic cone for elliptic surfaces with positive Euler number.
We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…
The paper proves an adjunction inequality for Real embedded surfaces in 4-manifolds.
problem Understanding the genus of Real embedded surfaces in 4-manifolds.
method Using equivariant cohomology and Real Seiberg-Witten invariants.
result Minimal genus of Real embedded surfaces can be larger than that of arbitrary embedded surfaces.
In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman …
We explain some interesting relations in the degree three bounded cohomology of surface groups. Specifically, we show that if two faithful Kleinian surface group representations are quasi-isometric, then their bounded fundamental classes are the same in bounded cohomology. This is novel in the setting that one end is d…
Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.
problem Classifying (2g+1)-dimensional complex linear representations of mapping class groups. method Using twisted 1-cohomology groups and Morita's computation, a complete classification is given for g≥7. result No irreducible linear representations of dimension 2g+1 for g≥7. Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the relationship between our classification and the second Poisson cohom…
We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of HdR1(S,R), or a single point. In the latter case, we show that S …
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of …
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For c…
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a 1st cohomology class of a 3-manifold the coefficients of twisted Alexander polynomials induce regular functions on the SL2(C)-character variety. We prove that if an ideal point giv…