Splitting criteria for homotopy functors of spectra
Phichet
Chaoha
1271-1280
Abstract: We explore the interaction between the Taylor tower and cotower, as defined in deriving calculus with cotriples and dual calculus for functors to spectra of functors of spectra. This leads to new splitting criteria which generalize the results in dual calculus for functors to spectra.
Random gaps under CH
James
Hirschorn
1281-1290
Abstract: It is proved that if the Continuum Hypothesis is true, then one random real always produces a destructible $(\omega_1,\omega_1)$ gap.
Involutions fixing ${\mathbb{RP}}^{\text{odd}} \sqcup P(h,i)$, II
Zhi
Lü
1291-1314
Abstract: This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space ${\mathbb{RP}}^j$ with its normal bundle nonbounding and a Dold manifold $P(h,i)$ with $h$ a positive even and $i>0$. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of $P(h,i)$ may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to $P(h,i)$. Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing ${\mathbb{RP}}^{\text{odd}}\sqcup P(h,i)$ is finished.
The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on $\mathcal{A}(\mathbb{R}^4)$
Rüdiger
W.
Braun;
Reinhold
Meise;
B.
A.
Taylor
1315-1383
Abstract: The local Phragmén-Lindelöf condition for analytic subvarieties of $\mathbb{C} ^n$ at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in $\mathbb{C} ^3$. The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on $\mathbb{R} ^4$.
There are no unexpected tunnel number one knots of genus one
Martin
Scharlemann
1385-1442
Abstract: We show that the only knots that are tunnel number one and genus one are those that are already known: $2$-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muñoz and by Morimoto and Sakuma. This confirms a conjecture first made by Goda and Teragaito.
Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs
Dmitri
Zaitsev;
Giuseppe
Zampieri
1443-1462
Abstract: The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of $\mathbb{C} ^{n}$. We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of BOGGESS-POLKING and EASTWOOD-GRAHAM.
LS-category of compact Hausdorff foliations
Hellen
Colman;
Steven
Hurder
1463-1487
Abstract: The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in $\{1,2, \ldots, \infty\}$. A foliation with all leaves compact and Hausdorff leaf space $M/\mathcal{F}$ is called compact Hausdorff. The transverse saturated category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}M$ of a compact Hausdorff foliation is always finite. In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(M)$ in terms of the geometry of $\mathcal{F}$ and the Epstein filtration of the exceptional set $\mathcal{E}$. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that \begin{displaymath}\max \{\operatorname{cat}(M/{\mathcal{F}}), \operatorname{ca... ...me{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(\mathcal{E}) + q.\end{displaymath} We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.
Analytic $p$-adic cell decomposition and integrals
Raf
Cluckers
1489-1499
Abstract: We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.
Stability of parabolic Harnack inequalities
Martin
T.
Barlow;
Richard
F.
Bass
1501-1533
Abstract: Let $(G,E)$ be a graph with weights $\{a_{xy}\}$ for which a parabolic Harnack inequality holds with space-time scaling exponent $\beta \ge 2$. Suppose $\{a_{xy}\}$. We prove that this parabolic Harnack inequality also holds for $(G,E)$ with the weights
Modular Shimura varieties and forgetful maps
Victor
Rotger
1535-1550
Abstract: In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.
Standard noncommuting and commuting dilations of commuting tuples
B.
V. Rajarama
Bhat;
Tirthankar
Bhattacharyya;
Santanu
Dey
1551-1568
Abstract: We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra $\mathcal{O}_n$ coming from dilations of commuting tuples.
Cohomology operations for Lie algebras
Grant
Cairns;
Barry
Jessup
1569-1583
Abstract: If $L$ is a Lie algebra over $\mathbb{R}$ and $Z$its centre, the natural inclusion $Z\hookrightarrow (L^{*})^{*}$ extends to a representation $i^{*}\,:\,\Lambda Z\to \operatorname{End} H^{*}(L,\mathbb{R})$ of the exterior algebra of $Z$ in the cohomology of $L$. We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of $\operatorname{End} H^{*}(L,\mathbb{R})$, and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to $0\to Z\to L\to L/Z\to 0$.
Copolarity of isometric actions
Claudio
Gorodski;
Carlos
Olmos;
Ruy
Tojeiro
1585-1608
Abstract: We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity $k$ representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.
Lattice invariants and the center of the generic division ring
Esther
Beneish
1609-1622
Abstract: Let $G$ be a finite group, let $M$ be a $ZG$-lattice, and let $F$ be a field of characteristic zero containing primitive $p^{{th}}$ roots of 1. Let $F(M)$ be the quotient field of the group algebra of the abelian group $M$. It is well known that if $M$ is quasi-permutation and $G$-faithful, then $F(M)^G$ is stably equivalent to $F(ZG)^G$. Let $C_n$ be the center of the division ring of $n\times n$ generic matrices over $F$. Let $S_n$ be the symmetric group on $n$symbols. Let $p$ be a prime. We show that there exist a split group extension $G'$of $S_p$ by a $p$-elementary group, a $G'$-faithful quasi-permutation $ZG'$-lattice $M$, and a one-cocycle $\alpha$ in
Overpartitions
Sylvie
Corteel;
Jeremy
Lovejoy
1623-1635
Abstract: We discuss a generalization of partitions, called overpartitions, which have proven useful in several combinatorial studies of basic hypergeometric series. After showing how a number of finite products occurring in $q$-series have natural interpretations in terms of overpartitions, we present an introduction to their rich structure as revealed by $q$-series identities.
Limit theorems for partially hyperbolic systems
Dmitry
Dolgopyat
1637-1689
Abstract: We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms. If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory.