On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

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 .

     

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.

     

Homomorphisms between Weyl modules for

We classify all homomorphisms between Weyl modules for when is an algebraically closed field of characteristic at least three, and show that the -spaces are all at most one dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Subgroups of acting transitively on -tuples

We consider subgroups of -diffeomorphisms of the circle which act transitively on -tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of . A stronger result concerning -approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.

     

On the mod cohomology of

We study the mod  cohomology of the classifying space of the projective unitary group . We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod  cohomology of the classifying space of connected compact Lie groups hold in the case of . Finally, we prove that the classifying space of the projective unitary group is determined by its mod  cohomology as an unstable algebra over the Steenrod algebra for 3$">, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases .

     

Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

-coarse geometry of complements of Z-sets in the Hilbert cube

Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube and the -coarse category of their complements. The -coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natural way some of the topological invariants of Z-sets to the geometry of their complements.

     

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.

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

On the -compact groups corresponding to the -adic reflection groups

There exists an infinite family of -compact groups whose Weyl groups correspond to the finite -adic pseudoreflection groups of family 2a in the Clark-Ewing list. In this paper we study these -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical -homomorphism. Finally, we also describe a faithful complexification homomorphism from these -compact groups to the -completion of unitary compact Lie groups.

     

-theorems for fusion systems

For an odd prime, we generalise the Glauberman-Thompson -nilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We define a notion of -free fusion systems and show that if is a -free fusion system on some finite -group , then is controlled by for any Glauberman functor , generalising Glauberman's -theorem (Glauberman, 1968) to arbitrary fusion systems.

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

Associahedra, cellular -construction and products of -algebras

The aim of this paper is to construct a functorial tensor product of -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.

     

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 .

     

Blakers-Massey elements and exotic diffeomorphisms of and via geodesics

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of and exotic diffeomorphisms of , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.