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.

     

Isomorphisms of function modules, and generalized approximation in modulus

For a function algebra we investigate relations between the following three topics: isomorphisms of singly generated -modules, Morita equivalence bimodules, and ``real harmonic functions' with respect to . We also consider certain groups which are naturally associated with a uniform algebra . We illustrate the notions considered with several examples.

     

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.

     

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.

     

Uniform and Lipschitz homotopy classes of maps

If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the -dimensional torus, it is bijective, while if is a compact orientable surface of genus 1$">, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that

     

Attractors for graph critical rational maps

We call a rational map graph critical if any critical point either belongs to an invariant finite graph , or has minimal limit set, or is non-recurrent and has limit set disjoint from . We prove that, for any conformal measure, either for almost every point of the Julia set its limit set coincides with , or for almost every point of its limit set coincides with the limit set of a critical point of .

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

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.

     

The Chromatic Ext Groups

We compute a certain Ext group related to the chromatic spectral sequence for , the spectrum whose -homology is for each . The answer we get displays a kind of periodicity not seen in the corresponding computation for the sphere spectrum.

     

Invariant ideals and polynomial forms

Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.

     

Euler characters and submanifolds of constant positive curvature

This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if is an -dimensional compact connected Riemannian submanifold of constant positive curvature in , then must be simply connected. It also gives a conformal version of this theorem.

     

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , and apply it to Lorentz and Orlicz sequence spaces.

     

Sums of squares in real analytic rings

Let be an analytic ring. We show: (1) has finite Pythagoras number if and only if its real dimension is , and (2) if every positive semidefinite element of is a sum of squares, then is real and has real dimension .

     

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 .

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

Differential operators on a polarized abelian variety

Let be an ample line bundle over a complex abelian variety . We show that the space of all global sections over of and are both of dimension one. Using this it is shown that the moduli space of rank one holomorphic connections on a compact Riemann surface does not admit any nonconstant algebraic function. On the other hand, is biholomorphic to the moduli space of characters of , which is an affine variety. So is algebraically distinct from the character variety if is of genus at least one.

     

Tilting theory and the finitistic dimension conjectures

Let be a right noetherian ring and let be the class of all finitely presented modules of finite projective dimension. We prove that findim iff there is an (infinitely generated) tilting module such that pd and . If is an artin algebra, then can be taken to be finitely generated iff is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.