Nonsimple, ribbon fibered knots
Katura
Miyazaki
1-44
Abstract: The connected sum of an arbitrary knot and its mirror image is a ribbon knot, however the converse is not necessarily true for all ribbon knots. We prove that the converse holds for any ribbon fibered knot which is a connected sum of iterated torus knots, knots with irreducible Alexander polynomials, or cables of such knots. This gives a practical method to detect nonribbon fibered knots. The proof uses a characterization of homotopically ribbon, fibered knots by their monodromies due to Casson and Gordon. We also study when cable fibered knots are ribbon and results which support the following conjecture. Conjecture. If a $(p,q)$ cable of a fibered knot $k$ is ribbon where $p(> 1)$ is the winding number of a cable in ${S^1} \times {D^2}$, then $ q = \pm 1$ and $ k$ is ribbon.
Topological applications of generic huge embeddings
Franklin D.
Tall
45-68
Abstract: In the Foreman-Laver model obtained by huge cardinal collapse, for many $ \Phi ,\Phi ({\aleph _1})$ implies $ \Phi ({\aleph _2})$. There are a variety of set-theoretic and topological applications, in particular to paracompactness. The key tools are generic huge embeddings and preservation via $\kappa$-centred forcing. We also formulate "potent axioms" à la Foreman which enable us to transfer from $ {\aleph _1}$ to all cardinals. One such axiom implies that all ${\aleph _1}$-collectionwise normal Moore spaces are metrizable. It also implies (as does Martin's Maximum) that a first countable generalized ordered space is hereditarily paracompact iff every subspace of size $ {\aleph _1}$ is paracompact.
Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras
Robert A.
Gustafson
69-119
Abstract: Multidimensional generalizations of beta type integrals of Barnes, Ramanujan, Askey-Wilson, and others are evaluated. These integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple Lie algebras of classical type and type $ {G_2}$. Many of these integrals can also be written as group integrals over a compact Lie group or conjugation invariant integrals over the corresponding Lie algebra.
Nonfibering spherical $3$-orbifolds
William D.
Dunbar
121-142
Abstract: Among the finite subgroups of $SO(4)$, members of exactly $21$ conjugacy classes act on $ {S^3}$ preserving no fibration of ${S^3}$ by circles. We identify the corresponding spherical $3$-orbifolds, i.e., for each such ${\mathbf{G}} < SO(4)$, we describe the embedded trivalent graph $\{ x \in {S^3}:\exists {\mathbf{I}} \ne {\mathbf{g}} \in {\mathbf{G}}$ s.t. ${\mathbf{g}}(x) = x\} /{\mathbf{G}}$ in the topological space $ {S^3}/{\mathbf{G}}$ (which turns out to be homeomorphic to $ {S^3}$ in all cases). Explicit fundamental domains (of Dirichlet type) are described for $9$ of the groups, together with the identifications to be made on the boundary. The remaining $ 12$ spherical orbifolds are obtained as mirror images or (branched) covers of these.
Co-Hopficity of Seifert-bundle groups
F.
González-Acuña;
R.
Litherland;
W.
Whitten
143-155
Abstract: A group $ G$ is cohopfian, if every monomorphism $G \to G$ is an automorphism. In this paper, we answer the cohopficity question for the fundamental groups of compact Seifert fiber spaces (or Seifert bundles, in the current vernacular). If $ M$ is a closed Seifert bundle, then the following are equivalent: (a) $ {\pi _1}M$ is cohopfian; (b) $M$ does not cover itself nontrivially; (c) $ M$ admits a geometric structure modeled on ${S^3}$ or on $ {\tilde{\text{SL}_2\mathbf{R}}}$. If $M$ is a compact Seifert bundle with nonempty boundary, then ${\pi _1}M$ is not cohopfian.
An explicit Plancherel formula for ${\rm U}(2,1)$
David
Jabon;
C. David
Keys;
Allen
Moy
157-171
Abstract: The admissible duals of quasi-split unitary groups over nonarchimedean fields are determined. The set of irreducible unitarizable representations, and the Plancherel measure on the unitary dual, is given explicitly.
The planar closing lemma for chain recurrence
Maria Lúcia Alvarenga
Peixoto;
Charles Chapman
Pugh
173-192
Abstract: If $z$ is a chain recurrent point of a $ {C^r}$ planar flow, all of whose fixed points are hyperbolic, then it is proved that the orbit through $z$ becomes periodic under a perturbation that is $ {C^r}$ small in the Whitney topology.
The structure of hyperfinite Borel equivalence relations
R.
Dougherty;
S.
Jackson;
A. S.
Kechris
193-225
Abstract: We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.
Contributions to the classification of simple modular Lie algebras
Georgia
Benkart;
J. Marshall
Osborn;
Helmut
Strade
227-252
Abstract: We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $ p > 7$. A $ 1$-section of such a Lie algebra relative to a torus $T$ of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum $ Q$ of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using $Q$ and obtain structural information about the $ 1$-sections and $ 2$-sections of $ Q$ relative to $ T$. We further classify all those algebras in which $Q$ is solvable.
The Brownian motion and the canonical stochastic flow on a symmetric space
Ming
Liao
253-274
Abstract: We study the limiting behavior of Brownian motion ${x_t}$ on a symmetric space $V = G/K$ of noncompact type and the asymptotic stability of the canonical stochastic flow ${F_t}$ on $O(V)$. We show that almost surely, $ {x_t}$ has a limiting direction as it goes to infinity. The study of the asymptotic stability of ${F_t}$ is reduced to the study of the limiting behavior of the adjoint action on the Lie algebra $\mathcal{G}$ of $G$ by the horizontal diffusion in $G$. We determine the Lyapunov exponents and the associated filtration of ${F_t}$ in terms of root space structure of $ \mathcal{G}$.
A functional from geometry with applications to discrepancy estimates and the Radon transform
Allen D.
Rogers
275-313
Abstract: Estimates of discrepancy, or irregularities of distribution, are obtained for measures without atoms. Two estimators are used, the half-space, or separation, discrepancy ${D_S}$ and a geometric functional ${I^\alpha }$. A representation formula for the generalized energy integral ${I^\alpha }$ is developed. Norm inequalities for the Radon transform are obtained as an application of the continuous discrepancy results. Integral geometric notions play a prominent role.
Representations of the symmetric group in deformations of the free Lie algebra
A. R.
Calderbank;
P.
Hanlon;
S.
Sundaram
315-333
Abstract: We consider, for a given complex parameter $\alpha$, the nonassociative product defined on the tensor algebra of $n$-dimensional complex vector space by $ [x,y] = x \otimes y - \alpha y \otimes x$. For $k$ symbols $ {x_1}, \ldots ,{x_k}$, the left-normed bracketing is defined recursively to be the bracketing sequence ${b_k}$, where $ {b_1} = {x_1}$, ${b_2} = [{x_1},{x_2}]$, and ${b_k} = [{b_{k - 1}},{x_k}]$. The linear subspace spanned by all multilinear left-normed bracketings of homogeneous degree $n$, in the basis vectors ${v_1}, \ldots ,{v_n}$ of ${\mathbb{C}^n}$, is then an ${S_n}$-module $ {V_n}(\alpha )$. Note that $ {V_n}(1)$ is the Lie representation $ \operatorname{Lie}_n$ of $ {S_n}$ afforded by the $ n$th-degree multilinear component of the free Lie algebra. Also, ${V_n}(- 1)$ is the subspace of simple Jordan products in the free associative algebra as studied by Robbins [Ro]. Among our preliminary results is the observation that when $\alpha$ is not a root of unity, the module ${V_n}(\alpha )$ is simply the regular representation. Thrall [T] showed that the regular representation of the symmetric group ${S_n}$ can be written as a direct sum of tensor products of symmetrised Lie modules ${V_\lambda }$. In this paper we determine the structure of the representations ${V_n}(\alpha )$ as a sum of a subset of these ${V_\lambda }$. The $ {V_\lambda }$, indexed by the partitions $\lambda$ of $n$, are defined as follows: let ${m_i}$ be the multiplicity of the part $ i$ in $\lambda$, let $\operatorname{Lie}_i$ be the Lie representation of $ {S_i}$, and let ${\iota _k}$ denote the trivial character of the symmetric group ${S_k}$. Let ${\iota _{{m_i}}}[\operatorname{Lie}_i]$ denote the character of the wreath product ${S_{{m_i}}}[{S_i}]$ of $ {S_{{m_i}}}$ acting on $ {m_i}$ copies of $ {S_i}$. Then ${V_\lambda }$ is isomorphic to the $ {S_n}$-module $\displaystyle ({\iota _{{m_1}}}[\operatorname{Lie}_1] \otimes \cdots \otimes {\... ...S_{m_1}}[{S_1}] \times \cdots \times {S_{{m_i}}}[{S_i}] \times \cdots }^{S_n}}.$ Our theorem now states that when $\alpha$ is a primitive $p$th root of unity, the ${S_n}$-module $ {V_n}(\alpha )$ is isomorphic to the direct sum of those ${V_\lambda }$, where $\lambda$ runs over all partitions $\lambda$ of $n$ such that no part of $\lambda$ is a multiple of $p$.
Polish group actions and the Vaught conjecture
Ramez L.
Sami
335-353
Abstract: We consider the topological Vaught conjecture: If a Polish group $ G$ acts continuously on a Polish space $S$, then $S$ has either countably many or perfectly many orbits. We show 1. The conjecture is true for Abelian groups. 2. The conjecture is true whenever $ G$, $S$ are recursively presented, the action of $G$ is recursive and, for $x \in S$ the orbit of $x$ is of Borel multiplicative rank $\leq \omega _1^x$. Assertion $1$ holds also for analytic $S$. Specializing $G$ to a closed subgroup of $ \omega !$, we prove that nonempty invariant Borel sets, not having perfectly many orbits, have orbits of about the same Borel rank. An upper bound is derived for the Borel rank of orbits when the set of orbits is finite.
Besov spaces on closed subsets of ${\bf R}\sp n$
Alf
Jonsson
355-370
Abstract: Motivated by the need in boundary value problems for partial differential equations, classical trace theorems characterize the trace to a subset $F$ of $ {\mathbb{R}^n}$ of Sobolev spaces and Besov spaces consisting of functions defined on $ {\mathbb{R}^n}$, if $ F$ is a linear subvariety $ {\mathbb{R}^d}$ of ${\mathbb{R}^n}$ or a $d$-dimensional smooth submanifold of ${\mathbb{R}^n}$. This was generalized in [2] to the case when $F$ is a $d$-dimensional fractal set of a certain type. In this paper, traces are described when $ F$ is an arbitrary closed set. The result may also be looked upon as a Whitney extension theorem in ${L^p}$.
The $H\sp p$-corona theorem for the polydisc
Kai-Ching
Lin
371-375
Abstract: Let ${H^p} = {H^p}({D^n})$ denote the usual Hardy spaces on the polydisc ${D^n}$. We prove in this paper the following theorem: Suppose ${f_1},{f_2}, \ldots ,{f_n} \in {H^\infty },{\left\Vert {{f_j}} \right\Vert _{{H^\infty }}} \leq 1$, and $ \sum\nolimits_{j = 1}^m {\vert{f_j}(z)\vert} \geq \delta > 0$. Then for every $g$ in ${H^p}$, $1 < p < \infty$, there are $ {H^p}$ functions $g,g, \ldots ,{g_m}$ such that $\sum\nolimits_{j = 1}^m {{f_j}(z){g_j}(z) = g(z)}$. Moreover, we have ${\left\Vert {{g_j}} \right\Vert _{{H^p}}} \leq c(m,n,\delta ,p){\left\Vert g \right\Vert _{{H^p}}}$. (When $p = 2,n = 1$, this theorem is known to be equivalent to Carleson's corona theorem.)
A weak characteristic pair for end-irreducible $3$-manifolds
Bobby Neal
Winters
377-403
Abstract: This extends a weakened version of the Characteristic Pair Theorem of Jaco, Shalen, and Johannson to a large subclass of the class of end-irreducible $ 3$-manifolds. The Main Theorem of this paper states that if $(W,w)$ is a noncompact $3$-manifold pair (where $W$ is a noncompact $3$-manifold that has an exhausting sequence with certain nice properties and where $ w$ is incompressible in $ W$), then there is a Seifert pair $(\Sigma ,\Phi )$ contained in $ (W,w)$ such that any $ 2$-manifold that is strongly essential in $(W,w)$ and each of whose components is a torus, an annulus, an open annulus, or a half-open annulus is isotopic in $(W,w)$ into $ (\Sigma ,\Phi )$.
Circular units of function fields
Frederick F.
Harrop
405-421
Abstract: A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.
Subgroup rigidity in finite-dimensional group algebras over $p$-groups
Gary
Thompson
423-447
Abstract: In 1986, Roggenkamp and Scott proved in [RS1] Theorem 1.1. Let $G$ be a finite $p$-group for some prime $p$, and $S$ a local or semilocal Dedekind domain of characteristic 0 with a unique maximal ideal containing $p$ (for example, $S = {\mathbb{Z}_p}$ where ${\mathbb{Z}_p}$ is the $p$-adic integers). If $ H$ is a subgroup of the normalized units of $SG$ with $\vert H\vert = \vert G\vert$, then $ H$ is conjugate to $ G$ by an inner automorphism of $SG$. In the Appendix of a later paper [S], Scott outlined a possible proof of a related result: Theorem 1.3. Let $S$ be a complete, discrete valuation domain of characteristic 0 having maximal ideal $ \wp$ and residue field $F \cong S/\wp $ of characteristic $ p$. Let $ G$ be a finite $ p$-group, and let $ U$ be a finite group of normalized units in $SG$. Then there is a unit $w$ in $SG$ such that $wU{w^{ - 1}} \leq G$. The author later filled in that outline to give a complete proof of Theorem 1.3 and, at the urging of Scott, has been able to extend that result to Theorem 1.2. Let $S$ be a complete, discrete valuation ring of characteristic 0 having maximal ideal $ \wp$ containing $ p$. Let $ A$ be a local $ S$-algebra that is finitely generated as an $S$-module, and let $G$ be a finite $p$-group. Then any finite, normalized subgroup of the $S$-algebra $\mathcal{A} = A{ \otimes _S}SG$ is conjugate to a subgroup of $G$.
On the rank and the crank modulo $4$ and $8$
Richard
Lewis;
Nicolas
Santa-Gadea
449-465
Abstract: In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo $4$ and $8$. These identities are similar to Dyson's identities for the rank modulo $5$ and $7$ which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.
Corrections to: ``First steps in descriptive theory of locales'' [Trans. Amer. Math. Soc. {\bf 327} (1991), no. 1, 353--371; MR1091230 (92b:54078)]
John
Isbell
467-468