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 .

     

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.

     

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.

     

Topological mixing in -spaces

If is a proper -space and a non-elementary discrete group of isometries acting properly discontinuously on it is shown that the geodesic flow on the quotient space is topologically mixing, provided that the generalized Busemann function has zeros on the boundary and the non-wandering set of the flow equals the whole quotient space of geodesics (the latter being redundant when is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete -spaces by a one-ended group of isometries and (C) finite -dimensional ideal polyhedra.

     

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.

     

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.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

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 .

     

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

     

Sufficient conditions for zero-one laws

We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size is bounded above by a polynomial in , then has a monadic second order - law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most is bounded above by a polynomial in , then this class has a first order - law. These results cover all known natural examples of classes of structures that have been proved to have a logical - law by Compton's method of analyzing generating functions.

     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .

     

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.

     

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.

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

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.

     

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.

     

Product systems over right-angled Artin semigroups

We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid . We show that the existing notions of product systems fit into our categorical framework, as do the -graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid . In particular, we obtain necessary and sufficient conditions under which a collection of -graphs form the coordinate graphs of a -graph.