Jumps in cohomology and free group actions

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if , then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.     

Finite group actions and étale cohomology

If a finite group G acts on a quasi-projective variety X, then H*_c(X,Z/n), the étale cohomology with compact support of X with coefficients inZ/n, has aZ/n[G]-module structure. It is well known that there is a finer invariant, an object RΓ_c(X,Z/n) of the derived category ofZ/n[G]-modules, whose cohomology is H*_c(X,Z/n). We show that there is a finer invariant still, a bounded complex Λ_c(X,Z/n) of direct summands of permutationZ/n[G]-modules, well-defined up to chain homotopy equivalence, which is isomorphic to RΓ_c(X,Z/n) in the derived category. This complex has many properties analogous to those of the simplicial chain complex of a simplicial complex with a group action. There are similar results forl-adic cohomology.     

On the cohomology of ergodic group actions

We consider three problems concerning cocycles of ergodic group actions: behavior of cohomology under the restriction of an ergodic semi-simple Lie group action to a lattice subgroup; “compactness” of bounded cocyles; and the relation of relative almost periodicity to relative discrete spectrum for extensions of ergodic actions.     

Quivers, Floer cohomology, and braid group actions

We consider the derived categories of modules over a certain family () of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type We show that each of these two rather different objects encodes the topology of curves on an -punctured disc. We prove that the braid group acts faithfully on the derived category of -modules, and that it injects into the symplectic mapping class group of the Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail yet). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of -modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

     

The cohomology of orbit spaces of certain free circle group actions

Suppose that \(G =\mathbb{S}^1\) acts freely on a finitistic space X whose (mod p) cohomology ring is isomorphic to that of a lens space \(L^{2m-1}(p;q_1,\ldots,q_m)\) or \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). The mod p index of the action is defined to be the largest integer n such that α ⁿ ?≠?0, where \(\alpha \,\epsilon\, H^2(X/G;\mathbb{Z}_p)\) is the nonzero characteristic class of the \(\mathbb{S}^1\)-bundle \(\mathbb{S}^1\hookrightarrow X\rightarrow X/G\). We show that the mod p index of a free action of G on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\) is p???1, when it is defined. Using this, we obtain a Borsuk–Ulam type theorem for a free G-action on \(\mathbb{S}^1\times \mathbb{C}P^{m-1}\). It is note worthy that the mod p index for free G-actions on the cohomology lens space is not defined.     

Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Duality and local group cohomology

Let be a group, let be a field, and let be a local system - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

Second bounded cohomology group

The goal of the paper is to prove that the canonical seminorm on the second bounded cohomology group is always a norm. The corresponding question remains open in the higher dimensional cases.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 117–120, 1988.     

Density of monodromy actions on non-abelian cohomology

In this paper we study the monodromy action on the first Betti and de Rham non-Abelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski-dense monodromy orbit. In particular, we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.     

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

Bounded cohomology of group constructions

It is proved that the singular partH _b ^2/(2) (G) of the second group of bounded homology of the discrete groupG is isomorphic to the space of 2-cocycles that vanish on the diagonal. For groupsG representable as HNN-extensions or free products with amalgamation, as well as for groupsG with one defining relation, conditions for the infinite-dimensionality ofH _b ^2/(2) (G) are found. Some applications of bounded cohomology to the width problem for verbal subgroups and to the boundedness problem for group presentations are indicated. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 546–550, April, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00239.     

Bredon-Illman cohomology with actions of totally disconnected groups

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.     

Toroidal orbifolds, gerbes and group cohomology

In this paper we compute the integral cohomology of certain semi-direct products of the form , arising from a linear action on the -torus, where is a finite group. The main application is the complete calculation of torsion gerbes for six-dimensional examples arising in string theory.