Model aspherical manifolds with no periodic maps

A. Borel proved that, if the fundamental group of an aspherical manifold is centerless and the outer automorphism group of is torsion-free, then admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on . In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented.

In this paper, we investigate to what extent the converse of Borel's result holds for aspherical manifolds arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion-free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.

     

Minimal isometric immersions of inhomogeneous spherical space forms into spheres--- a necessary condition for existence

Although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms, there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, we give a necessary condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. Evaluating this function shows that neither nor admit a minimal isometric immersion into any sphere if the degree of the immersion is less than , or less than , respectively.

     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

The Witt classes of torsion linking forms of (4n−1)-manifolds with pseudo-free circle actions

In this paper we compute the Witt class of torsion linking forms of (4n−1)-dimensional closed oriented manifolds admitting orientation preserving pseudo-free circle actions.     

On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53-64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119-171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173-177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91-117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17-22, 2001, TIFR, Mumbai, 2001, pp. 469-490] of D. Morris (Witte) on a completely different topic. We will answer that question too.     

Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity

In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series. Copyright © 2015 John Wiley & Sons, Ltd.     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

Topological,affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π₀(Homeo(M))→Out π₁(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.     

Conjugacy problem in groups of oriented geometrizable 3-manifolds

The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u,v of π₁(M) whether u and v are in the same conjugacy class of π₁(M) or not. More topologically, the algorithm decides for any two loops in M, whether they are freely homotopic or not.     

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group and a real form we define a Poisson structure on the variety of Borel subgroups of with the property that all -orbits in as well as all Bruhat cells (for a suitable choice of a Borel subgroup of ) are Poisson submanifolds. In particular, we show that every non-empty intersection of a -orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

     

Realizing representations on generalized flag manifolds

Let G be a complex reductive linear algebraic group and a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that is a real form of L.Using the minimal globalization V _min of a finite length admissible representation for L ₀, one can define a homogeneous analytic vector bundle on the G ₀ orbit S of P in the generalized flag manifold . Let denote the corresponding sheaf of polarized sections. In this article we analyze the G ₀ representations obtained on the compactly supported sheaf cohomology groups .     

Strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms

Abstract

This article is devoted to C² a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.     

On K 2-group of a formal matrix ring

Let S be a formal matrix ring, T the subring consisting of all diagonal elements, I the set consisting of all off-diagonal elements. Then I is a split radical ideal under certain conditions. In this paper, we show that K2 (S)■K2 (T) ? K2 (S,I), and a presentation of K2 (S,I) is given.     

Semiglobal results for on a complex space with arbitrary singularities

We obtain some -results for the operator on forms that vanish to high order on the singular set of a complex space.

     

On compact symplectic manifolds with Lie group symmetries

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.

     

Construction of invariant scalar particle wave equations on Riemannian manifolds with external gauge fields

We consider the problem of constructing scalar particle wave equations in Riemannian spaces with external gauge fields whose symmetry group is the group of motions of the Riemannian space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 237–249, August, 2008.     

Obstructions to splitting manifolds with infinite fundamental group

In this paper we calculate the obstruction groups to splitting along one-sided submanifolds when the fundamental group of the submanifold is isomorphic to or /2. We also consider the case where the obstruction group is not a Browder-Livesey group. We construct a new Levine braid that connects the Wall groups to the obstruction group for splitting. We solve the problem of the mutual disposition of images of several natural maps in Wall groups for finite 2-groups with exceptional orientation character.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 163–175, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.