New abelian cohomology theory applied to rigidity of flows.
problem Rigidity of Anosov flows and negatively curved surfaces.
method Development of abelian Livshits theory for transitive Anosov flows.
result Abelian Livshits theorem for homologically full Anosov flows.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
Relations between parameter rigidity of locally free Lie group actions on closed manifolds and the 1st leafwise cohomology of the orbit foliations are discussed. Some computational results of the leafwise cohomology are included.
Counterexample shows ADC contact structures can't have isomorphic cohomologies.
problem Rigidity of ADC contact structures and cohomology isomorphisms.
method Provided a counterexample to show non-isomorphic cohomologies.
result ADC contact structures do not have isomorphic integral cohomologies.
In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the L…
Characterizes rigidity of compact foliations using deformations of Lie groupoids and algebroids.
problem Rigidity of compact foliations on manifolds.
method Combining stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids.
result Cohomological characterization for rigidity of compact foliations.
The study classifies nilpotent Lie foliations with cohomological obstructions.
problem Understanding rigidity of nilpotent Lie foliations under solvable deformations.
method Development of a cohomological framework and algebraic criterion for rigidity.
result Established a necessary and sufficient algebraic criterion for rigidity in generalized Heisenberg groups.
New cohomological rigidity results for manifolds defined by right-angled polytopes.
problem Establishing cohomological rigidity for manifolds defined by specific polytopes.
method Using techniques from toric topology, the authors prove cohomological rigidity for families of manifolds associated with polytopes from a specific class.
result Cohomology ring isomorphisms imply diffeomorphisms for manifolds in the families, and vice versa.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
A complex projective tower or simply a CP-tower is an iterated complex projective fibrations starting from a point. In this paper we classify all 6-dimensional CP-towers up to diffeomorphism, and as a consequence, we show that all such manifolds are cohomologically rigid, i.e., they are completely d…
Elliptic bouquets defined for spin manifolds with circular actions.
problem Defining integration in elliptic cohomology.
method Introducing elliptic bouquets of germs of holomorphic equivariant cohomology classes, integrating them as in K-theory.
result Witten's rigidity theorem follows from integration of elliptic bouquets.
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology.
problem Cohomology of the Regge complex in three dimensions.
method Constructing a discrete version of linearized Riemann-Cartan geometry on any triangulation.
result The cohomology of the Regge complex is isomorphic to the infinitesimal-rigid-body-motion-valued de~Rham cohomology.
New metric on cohomology space shows flexibility and rigidity.
problem Metric geometry of big cohomology classes.
method Introduced a new metric d1 on finite energy space E1(X,θ). result Geodesic rays can be constructed in the space.
When $X=Γ\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.
New manifold examples found in higher dimensions.
problem Cohomological rigidity and chromatic numbers of polytopes.
method Investigation of small covers and quasitoric manifolds over polytopes.
result Found new examples of quasitoric manifolds with specific chromatic numbers.
The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.
problem Understanding the topological and geometric properties of foliations.
method Applications of higher Lefschetz theorems for foliations, involving Haefliger cohomology.
result The non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions.
The paper proves vanishing cohomology groups for free boundary hypersurfaces.
problem Proving vanishing cohomology groups for free boundary hypersurfaces.
method Using a universal constant and traceless second fundamental form condition.
result The pth cohomology group of a compact free boundary submanifold vanishes. Killing-Yano and conformal Killing-Yano superalgebras are rigid in constant curvature manifolds.
problem Understanding the rigidity of Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds.
method Defining Z-gradations and filtrations, showing trivial second cohomology groups, and proving non-deformability. result Killing-Yano and conformal Killing-Yano superalgebras are rigid and correspond to geometric invariants of constant curvature manifolds.
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.
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…
We use cross ratios to describe second real continuous bounded cohomology for locally compact topological groups. We also derive a rigidity result for cocycles with values in the isometry group of a proper hyperbolic geodesic metric space.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance…
New proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.
problem Determining if 4D symplectic manifolds are diffeomorphic based on their equivariant cohomology.
method Proved that equivariant cohomology rings of Hamiltonian circle actions on 4D symplectic manifolds determine their equivariant diffeotypes.
result Isomorphism of equivariant cohomology rings implies equivariant diffeomorphism for 4D symplectic manifolds.
Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
The paper explores higher property T in lattices and its connections to geometric phenomena.
problem Understanding higher property T in lattices and related geometric phenomena.
method Operator-algebraic characterizations of higher property T and connections to lattice geometry.
result Unified framework for understanding higher property T and related geometric phenomena.
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.
problem Computing the cohomology of mapping class groups with level structures and Prym representations.
method Using twisted cohomology and Prym representations for any positive integer r.
result Cohomology exhibits instability for large genus, but remains stable for r=0 or r=1.
In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) \cite[Chapter 8, §2 Gluing Manifolds Together]{h}, we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimension…
We prove a general extrinsic rigidity theorem for homogeneous varieties in CPN. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in Pg) is extrinsically rigid to third order. In contrast, we show that the ad…
There is an error in the proof of Proposition 3.7. Proposition 3.7 is needed for the proof of the main theorem.
There is an error in the proof of Theorem 1.1 that invalidates proofs of other theorems. Theorem 1.5 is unaffected.
We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature κ∈{−1,0,1} and cone-angles ≤π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem fo…
For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using tech…
We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the Farb-Kaimanovich-Masur rigidity theorem that states that MCG(S) does not contain a higher rank lattic…
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SLn(Z) at virtual cohomological dimension. method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
We prove that the second Hochschild cohomology group of the moduli stack of stable n-pointed genus g curves vanishes for all but finitely many (g,n).
In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees (2p−2)-cohomology with $\C/\Z$-cohomology, for p≥2. Furthermore, they are shown to be rigid in a variation of paths (p…
This paper extends Witten's holomorphic Morse inequalities to singular spaces.
problem Applying Witten's holomorphic Morse inequalities to singular spaces.
method Constructing Witten instanton complexes for Kähler Hamiltonian Morse functions on stratified pseudomanifolds.
result Extends Witten's holomorphic Morse inequalities to singular spaces.
Study calculates volume variations for hyperbolic 3-manifolds.
problem Calculating volume variations for hyperbolic 3-manifolds.
method Uses Lie algebra cohomology and variation strategy similar to Reznikov's rigidity.
result Derives a variation formula for the volume of representations in SL_n(C).
We prove that if Γ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of Γ equals its proper geometric dimension.
Metrics are isometric for certain Anosov magnetic systems.
problem Isometry of metrics for Anosov magnetic systems.
method Conjugacy isotopic to the identity, volume-preserving conjugacy, cohomology class.
result Isometric metrics for conjugate Anosov magnetic systems.
Spin-structures on real Bott manifolds with Kähler structures are characterized.
problem Existence of Spin-structures on real Bott manifolds with Kähler structures.
method Ishida characterization and Popko-Szczepański cohomological rigidity technique.
result Necessary and sufficient condition for the existence of Spin-structures on M.
Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL3(R) and 5-dimensional solvable Lie groups, use spectral sequence arguments. result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.
Survey explores cohomology's roles in applied math and sciences.
problem Understanding cohomology's role in solving differential equations.
method Examining differential complexes and structure-preserving discretizations.
result Various fundamental concepts in mechanics are formulated using differential complexes.
We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary r…
These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (jo…
The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.
problem Certifying infinitesimal projective rigidity for hyperbolic once-punctured torus bundles.
method Using twisted Alexander polynomials of representations associated with the holonomy.
result The induced action on the tangent space of the character variety matches the group theoretic action.