Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

Local cohomology based on a nonclosed support defined by a pair of ideals

We introduce a generalization of the notion of local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I,J), and study its various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with ordinary local cohomology.     

\boldsymbol{\mathfrak{F}}-Structures and Bredon–Galois Cohomology

Let $\mathfrak{F}$ be an arbitrary family of subgroups of a group G and let $\mathcal{O}_\mathfrak{F}G$ be the associated orbit category. We investigate interpretations of low dimensional $\mathfrak{F}$ -Bredon cohomology of G in terms of abelian extensions of $\mathcal{O}_\mathfrak{F}G$ . Specializing to fixed point functors as coefficients, we derive several group theoretic applications and introduce Bredon–Galois cohomology. We prove an analog of Hilbert’s Theorem 90 and show that the second Bredon–Galois cohomology is a certain intersection of relative Brauer groups. As applications, we realize the relative Brauer group Br(L/K) of a finite separable non-normal extension of fields L/K as a second Bredon cohomology group and show that this approach is quite suitable for finding nonzero elements in Br(L/K).     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

Finite Groups Acting on Homology 3-Spheres: On a Result of M. Reni

 We give a list including all finite groups G which admit smooth orientation preserving actions on homology 3-spheres (arbitrary actions, i.e. possibly with fixed points; if the action is free then the group G has periodic cohomology and the classification of such groups is well known). The main work in this direction is due to M. Reni. In the present paper, we complete and extend his results for the case of nonsolvable groups G. Received 19 March 2001; in revised form 15 September 2001     

Boundaries of right-angled Coxeter groups with manifold nerves

All abstract reflection groups act geometrically on non-positively curved geodesic spaces. Their natural space at infinity, consisting of (bifurcating) infinite geodesic rays emanating from a fixed base point, is called a boundary of the group.We will present a condition on right-angled Coxeter groups under which they have topologically homogeneous boundaries. The condition is that they have a nerve which is a connected closed orientable PL manifold.In the event that the group is generated by the reflections of one of Davis’ exotic open contractible n-manifolds (n?4), the group will have a boundary which is a homogeneous cohomology manifold. This group boundary can then be used to equivariantly Z-compactify the Davis manifold.If the compactified manifold is doubled along the group boundary, one obtains a sphere if n?5. The system of reflections extends naturally to this sphere and can be augmented by a reflection whose fixed point set is the group boundary. It will be shown that the fixed point set of each extended original reflection on the thus formed sphere is a tame codimension-one sphere.     

Butterflies II: Torsors for 2-group stacks

We study torsors over 2-groups and their morphisms. In particular, we study the first non-abelian cohomology group with values in a 2-group. Butterfly diagrams encode morphisms of 2-groups and we employ them to examine the functorial behavior of non-abelian cohomology under change of coefficients. We re-interpret the first non-abelian cohomology with coefficients in a 2-group in terms of gerbes bound by a crossed module. Our main result is to provide a geometric version of the change of coefficients map by lifting a gerbe along the “fraction” (weak morphism) determined by a butterfly. As a practical byproduct, we show how butterflies can be used to obtain explicit maps at the cocycle level. In addition, we discuss various commutativity conditions on cohomology induced by various degrees of commutativity on the coefficient 2-groups, as well as specific features pertaining to group extensions.     

Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ₁ and φ₂ of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [M,φ₁]=[M,φ₂] in the oriented bordism group Ω₂(G) of the group G. In this paper, we compute the bordism class [M,φ] for any such action of G on M and we determine for a given M, the bordism classes in Ω₂(G) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form n(Z/pαZ)).     

Equivariant cohomology of K-contact manifolds

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, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

A fixed point theorem for exponentially bounded groups

A fixed point property for linear actions of locally compact groups is presented. It is shown, in particular, that if G is either a finitely generated, discrete solvable group or a connected group then G has this fixed point property if, and only if, G is exponentially bounded.     

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.     

De Rham cohomology of manifolds containing the points of infinite type,and Sobolev estimates for the\bar \partial - Neumann problem

We consider smooth bounded pseudoconvex domains Ω in Cⁿ whose boundary points of infinite type are contained in a smooth submanifoldM (with or without boundary) of the boundary having its (real) tangent space at each point contained in the null space of the Levi form ofbΩ at the point. (In particular, complex submanifolds satisfy this condition.) We consider a certain one-form α onbΩ and show that it represents a De Rham cohomology class on submanifolds of the kind described. We prove that if α represents the trivial cohomology class onM, then the Bergman projection and the \(\bar \partial - Neumann\) operator on Ω are continuous in Sobolev norms. This happens, in particular, ifM has trivial first De Rham cohomology, for instance, ifM is simply connected.     

Relative Projectivity of Modules and Cohomology Theory of Finite Groups

In the modular representation theory of finite groups the theory of projectivity relative to subgroups is of fundamental importance. To generalize this notion the theory of projectivity relative to modules was introduced by the first author. Our aim is to show some aspects of cohomology theory of finite groups concerning projectivity of modules relative to both subgroups and modules. We shall give some applications to the cohomology theory; especially we shall calculate the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups.     

Group actions and Helly's theorem

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encoded in a property that we introduce called “property FAr”, which reduces to Serre's property FA when r=1. The method applies to S-arithmetic groups in higher Q-rank, to simplex reflection groups (including some nonarithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example SLn(Z[x₁,…,xd]), n>2).     

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.     

On purely non-free finite actions of abelian groups on compact surfaces

A finite group of self-homeomorphisms of a closed orientable surface is said to act on it purely non-freely if each of its elements has a fixed point; we also call it a gpnf-action. In this paper we observe that gpnf-actions exist for an arbitrary finite group and we discuss the minimum genus problem for such actions. We solve it for abelian groups. In the cyclic case we prove that the minimal gpnf-action genus coincides with Harvey’s minimal genus.     

Differential conformal superalgebras and their forms

We introduce the formalism of differential conformal superalgebras, which we show leads to the “correct” automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N=2 and N=4 conformal superalgebras by means of Galois cohomology.