Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

On the Glauberman and Watanabe correspondences for blocks of finite -solvable groups

If is a finite -solvable group for some prime , a solvable subgroup of the automorphism group of of order prime to such that stabilises a -block of and acts trivially on a defect group of , then there is a Morita equivalence between the block and its Watanabe correspondent of , given by a bimodule with vertex and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).

     

Milnor classes of local complete intersections

Let be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component of the singular set of , we define the Milnor class in the homology of . The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of is shown to be equal to the sum of over the connected components of . This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The -degree component coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for when is a non-singular component and satisfies the Whitney condition along .

     

The extraspecial case of the problem

Let be an extraspecial-type group and a faithful, absolutely irreducible -module, where is a finite field. Let be the normalizer in of . We show that, with few exceptions, there exists a such that the restriction of to is self-dual whenever and .

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch -equivalent to the unit vector basis of , then for all 0$">, there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.

     

Induced operators on symmetry classes of tensors

Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and is a character of degree 1 on . Consider the symmetrizer on the tensor space

defined by and . The vector space

is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying

In this paper, several basic problems on induced operators are studied.

     

On the blow-up of heat flow for conformal -harmonic maps

Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from (a unit ball in ) to (a unit sphere in ) under certain initial and boundary conditions. We generalize this result to the case of -harmonic map heat flow from to . In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.

     

Weakly defective varieties

A projective variety is `-weakly defective' when its intersection with a general -tangent hyperplane has no isolated singularities at the points of tangency. If is -defective, i.e. if the -secant variety of has dimension smaller than expected, then is also -weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of -defective surfaces.

     

Spin structures and codimension two embeddings of -manifolds up to regular homotopy

We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into -space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of -manifolds , the following phenomenon occurs: there exists a codimension two immersion of the -sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of . This involves introducing and studying an equivalence relation on the set of spin structures on . Their associated -invariants also play an important role.

     

New bases for Triebel-Lizorkin and Besov spaces

We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the , , potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function . Typical examples of such 's are the rational function and the Gaussian function This paper also shows how the new bases can be utilized in nonlinear approximation.

     

Detection of renewal system factors via the Conley index

Let be an isolating neighborhood for a map . If we can decompose into the disjoint union of compact sets and , then we can relate the dynamics on the maximal invariant set to the shift on two symbols by noting which component of each iterate of a point lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to . In essence, we measure the difference between the Conley index of and the sum of the indices of and .

     

Lower central series and free resolutions of hyperplane arrangements

If is the complement of a hyperplane arrangement, and is the cohomology ring of over a field of characteristic , then the ranks, , of the lower central series quotients of can be computed from the Betti numbers, , of the linear strand in a minimal free resolution of over . We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, , of a minimal resolution of over the exterior algebra .

From this analysis, we recover a formula of Falk for , and obtain a new formula for . The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra is Koszul if and only if the arrangement is supersolvable.

We also give combinatorial lower bounds on the Betti numbers, , of the linear strand of the free resolution of over ; if the lower bound is attained for , then it is attained for all . For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that is determined by the number of triangles and subgraphs in the graph.

     

Small rational model of subspace complement

This paper concerns the rational cohomology ring of the complement of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for . Inside it we find a much smaller subalgebra quasi-isomorphic to the whole algebra. is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of . The algebra has a natural integral version that is a good candidate for an integral model of . If the rational local homology of can be computed explicitly we obtain an explicit presentation of the ring . For example, this is done for the cases where is a geometric lattice and where is a -equal manifold.

     

Self-tilting complexes yield unstable modules

Let be a group and a commutative ring. Let be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of -modules. The subgroup of consisting of self-equivalences fixing the trivial -module acts on the cohomology ring . The action is functorial with respect to . The self-equivalences which are 'splendid' in a sense defined by J. Rickard act naturally with respect to transfer and restriction to centralizers of -subgroups in case is a field of characteristic . In the present paper we prove that this action of self-equivalences on commutes with the action of the Steenrod algebra, and study the behaviour of the action of splendid self-equivalences with respect to Lannes' -functor.

     

Katetov's problem

In 1948 Miroslav Katetov showed that if the cube of a compact space satisfies the separation axiom T then must be metrizable. He asked whether can be replaced by in this metrization result. In this note we prove the consistency of this implication.

     

Character degrees and nilpotence class of finite groups

Let be a finite set of powers of containing 1. It is known that for some choices of , if is a finite -group whose set of character degrees is , then the nilpotence class of is bounded by some integer that depends on , while for some other choices of such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set is class bounding if and only if . In this article we provide a new approach to this problem. Our main result shows the relevance of certain -adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets such that .

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Finely -harmonic functions of a Markov process

Let be a Borel right process and a fixed excessive measure. Given a finely open nearly Borel set we define an operator which we regard as an extension of the restriction to of the generator of . It maps functions on to (locally) signed measures on not charging -semipolars. Given a locally smooth signed measure we define to be (finely) -harmonic on provided on and denote the class of such by . Under mild conditions on we show that is equivalent to a local ``Poisson' representation of . We characterize by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.