Nullification and cellularization of classifying spaces of finite groups

In this note we discuss the effect of the -nullification and the -cellularization over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe by means of a covering fibration, and we classify all finite groups for which is -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of and

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

Examples for the mod motivic cohomology of classifying spaces

Let be the classifying space of a compact Lie group . Some examples of computations of the motivic cohomology are given, by comparing with , and .

     

Scrollar syzygies of general canonical curves with genus

We prove that for a general canonical curve of genus , the space of th (last) scrollar syzygies is isomorphic to the Brill-Noether locus . Schreyer has conjectured that these scrollar syzygies span the space of all th (last) syzygies of . Using Mukai varieties we prove this conjecture for genus , and .

     

Lusternik-Schnirelmann category of

First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.

     

Iwasawa theory for -local spectra

The Iwasawa algebra is a power series ring in one variable over the -adic integers. It has long been studied by number theorists in the context of -extensions of number fields. It also arises, however, as a ring of operations in -adic topological -theory. In this paper we study -local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for odd we classify -local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for -modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

     

Values of Gauss maps of complete minimal surfaces in on annular ends

Let be a complete minimal surface in and let be an annular end of which is conformal to , where is the conformal coordinate. Let be the generalized Gauss map of . We show that must intersect every hyperplane in , with the possible exception of hyperplanes in general position.

     

All -local finite groups of rank two for odd prime

In this paper we give a classification of the rank two -local finite groups for odd . This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible -radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic -local finite group, getting some new examples of these for .

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

Every real ellipsoid in admits CR umbilical points

We prove that every real ellipsoid admits at least four umbilical points, which can be compared to the result of Webster that a generic real ellipsoid in with does not admit any umbilical point.

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .

     

Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

     

Compatible valuations and generalized Milnor -theory

Given a field and a subgroup of there is a minimal group for which there exists an -compatible valuation whose units are contained in . Assuming that has finite index in and contains for prime, we describe in computable -theoretic terms.

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

Nonabelian cohomology with coefficients in Lie groups

In this paper we prove some properties of the nonabelian cohomology of a group with coefficients in a connected Lie group . When is finite, we show that for every -submodule of which is a maximal compact subgroup of , the canonical map is bijective. In this case we also show that is always finite. When and is compact, we show that for every maximal torus of the identity component of the group of invariants , is surjective if and only if the -action on is -semisimple, which is also equivalent to the fact that all fibers of are finite. When , we show that is always surjective, where is a maximal compact torus of the identity component of . When is cyclic, we also interpret some properties of in terms of twisted conjugate actions of .

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

On -adic intermediate Jacobians

For an algebraic variety of dimension with totally degenerate reduction over a -adic field (definition recalled below) and an integer with , we define a rigid analytic torus together with an Abel-Jacobi mapping to it from the Chow group of codimension algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over . We compare and contrast the complex and -adic theories. Finally, we examine a special case of a -adic analogue of the Generalized Hodge Conjecture.

     

Extensions of -local finite groups

A -local finite group consists of a finite -group , together with a pair of categories which encode ``conjugacy' relations among subgroups of , and which are modelled on the fusion in a Sylow -subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as -completed classifying spaces of finite groups. In this paper, we study and classify extensions of -local finite groups, and also compute the fundamental group of the classifying space of a -local finite group.

     

The arithmetic and combinatorics of buildings for

In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for with a local field. We characterize the action of the affine Weyl group in terms of a symplectic basis for an apartment, characterize the special vertices as those which are self-dual with respect to the induced inner product, and establish a one-to-one correspondence between the special vertices in an apartment and the elements of the quotient .

We then give a natural representation of the local Hecke algebra over acting on the special vertices of the Bruhat-Tits building for . Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for .