arXiv research

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.

Study confirms optimal bounds for group cohomology of Lie groups.

Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.

Study finite group actions on 4-manifolds, finding rank bounds.

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency d…

The study proves stability of a flow on specific Lie groups.

Improved bounds on $\Z_2$-torus actions on positively curved manifolds.

Study shows non-trivial homology classes in certain symmetric manifolds.

The paper disproves a generalized toral rank conjecture with various counter-examples.

The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. …

We discuss the Morse-Novikov cohomology of a compact manifold, associated to a closed one--form whose free abelian group generated by its periods $\langle \int_γη\mid [γ] \in π_1(M)\rangle$ is of rank 1, the focus being on locally conformally symplectic manifolds. In particular, we provide an explicit computation for t…

Study Betti and Hodge numbers of solvmanifolds from integer polynomials.

The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We det…

$L^p$-cohomology of rank one symmetric spaces of noncompact type is shown to be Hausdorff for values of $p$ where this does not follow from curvature pinching. Using the multiplicative structure on $L^p$-cohomology, it is shown that no simply connected Riemannian manifold with strictly -1/4-pinched sectional curvature …

We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…

We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah-Bott asserts that its rational cohomology ring is generated by the universal clas…

Paper investigates double coverings and torsions in homology groups.

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 $Σ\times S^1$. (Note: There is some…

Study manifolds with positive intermediate Ricci curvature and large symmetry rank.

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

Complex group cohomology surprisingly simple.

The paper bounds Betti numbers of complex-hyperbolic manifolds.

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $\mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-…

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is…

We propose a new method for studying $n$- and $Γ$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $Γ$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $Γ$-cohomology of…

Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.

Study of Eisenstein series linked to hyperbolic cusps.

Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.

Let $\A$ be a line arrangement in the complex projective plane $\PP^2$. Denote by $M$ its complement and by $\M$ the set of points in $\A$ with multiplicity at least 3. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the dimension of the cohomology groups $H^m(M,\mathcal{L})$ can be compu…

We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symm…

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.

T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford-Kl…

Proof of boundedness of quasimorphisms for certain Lie groups.

Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.

The paper explores higher property T in lattices and its connections to geometric phenomena.

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class,…

In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish…

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

We survey generalisations of the Chang-Skjelbred Lemma for integral coefficients. Moreover, we construct examples of manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday's. The "mutants" we construct are obtained from compactified repre…

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…

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp…

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 $Σ\times P^1$. We discuss on to how extent the Quantum cohomology of M determines its Gromow-Witten invariants. …

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce …

The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the modul…