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European Mathematical Society Publishing House
2024-03-29 07:11:40
10
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=79&iss=2&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
79
2004
2
Une version feuilletée du théorème de translation de Brouwer
Patrice
Le Calvez
Université de Paris XIII/CNRS, VILLETANEUSE, FRANCE
Homéomorphisme du plan, droite de Brouwer, feuilletage
The Brouwer's plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a proper topological imbedding C of R, disjoint from its image and separating $f(C)$ and $f^{-1}(C)$. Such a curve is called a Brouwer line. We prove that we can construct a foliation of the plane by Brouwer lines.
Dynamical systems and ergodic theory
General
229
259
10.1007/s00014-003-0745-9
http://www.ems-ph.org/doi/10.1007/s00014-003-0745-9
Integral bases for TQFT modules and unimodular representations of mapping class groups
Patrick
Gilmer
Louisiana State University, BATON ROUGE, UNITED STATES
Gregor
Masbaum
Université Paris 7, Denis Diderot, PARIS CEDEX 05, FRANCE
Paul
van Wamelen
Louisiana State University, BATON ROUGE, UNITED STATES
Topological quantum field theory, mapping class group, integrality
We construct integral bases for the $SO(3)$-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be non-unimodular. In one such case, genus three at a fifth root of unity, we still give an explicit basis.
Manifolds and cell complexes
General
260
284
10.1007/s00014-004-0801-5
http://www.ems-ph.org/doi/10.1007/s00014-004-0801-5
Realizing connected Lie groups as automorphism groups of complex manifolds
Jörg
Winkelmann
Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE
Lie groups, automorphism groups, Stein hyperbolic complex manifolds, bounded domains
We show that every connected real Lie group can be realized as the full automorphism group of a Stein hyperbolic complex manifold.
Several complex variables and analytic spaces
Topological groups, Lie groups
General
285
299
10.1007/s00014-003-0794-5
http://www.ems-ph.org/doi/10.1007/s00014-003-0794-5
On the volume of unit vector fields on spaces of constant sectional curvature
Fabiano
Brito
Universidade de São Paulo, SÃO PAULO SP, BRAZIL
Pablo
Chacón
Universidad de Murcia, MURCIA, SPAIN
A.
Naveira
Universidad de Valencia, BURJASSOT, SPAIN
Unit vector fields, Sasaki metric, volume functional, spaces forms
A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.
Differential geometry
Global analysis, analysis on manifolds
General
300
316
10.1007/s00014-004-0802-4
http://www.ems-ph.org/doi/10.1007/s00014-004-0802-4
Families of strong KT structures in six dimensions
Anna
Fino
Università degli Studi di Torino, TORINO, ITALY
Maurizio
Parton
Università di Chieti-Pescara, PESCARA, ITALY
Simon
Salamon
King's College London, LONDON, UNITED KINGDOM
Hermitian metric, complex structure, connection, torsion, nilmanifold
This paper classifies Hermitian structures on 6-dimensional nilmanifolds $M=\Ga\bs G$ for which the fundamental 2-form is $\pd\opd$-closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the complex plane. Related theory is used to provide examples of various types of Ricci-flat structures.
Differential geometry
Nonassociative rings and algebras
Several complex variables and analytic spaces
Quantum theory
317
340
10.1007/s00014-004-0803-3
http://www.ems-ph.org/doi/10.1007/s00014-004-0803-3
The stable equivalence and cancellation problems
Leonid
Makar-Limanov
Wayne State University, DETROIT, UNITED STATES
Peter
van Rossum
New Mexico State University, LAS CRUCES, UNITED STATES
Vladimir
Shpilrain
City University of New York, NEW YORK, UNITED STATES
JIE-TAI
YU
UNIVERSITY OF HONG KONG, HONG KONG, CHINA
Algebraic varieties, cancellation problem, polynomial automorphisms, stable equivalence, Danielewski surfaces
Let K be an arbitrary field of characteristic 0, and $\mathbf{A}^n$ the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphic cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether $V_1$ and $V_2$ themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$) cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are $V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$? We call this stable equivalence problem. We show that the answer is positive for any two curves $V_1, V_2 \subseteq \mathbf{A}^2$. For an arbitrary $n \ge 2$, we consider a special, arguably the most important, case of both problems, where one of the varieties is a hyperplane. We show that a positive solution of the stable equivalence problem in this case implies a positive solution of the cancellation problem.
Algebraic geometry
Commutative rings and algebras
General
341
349
10.1007/s00014-003-0796-3
http://www.ems-ph.org/doi/10.1007/s00014-003-0796-3
On the zero set of semi-invariants for tame quivers
Christine
Riedtmann
Universität Bern, BERN, SWITZERLAND
Grzegorz
Zwara
Nicolaus Copernicus University, TORUŃ, POLAND
Semi-invariants, quivers, representations
Let d be a prehomogeneous dimension vector for a finite tame quiver Q. We show that the common zeros of all non-constant semi-invariants for the variety of representations of Q with dimension vector $N\cdot\mathbf d$, under the product of the general linear groups at all vertices, is a complete intersection for $N\geq 3$.
Algebraic geometry
Associative rings and algebras
General
350
361
10.1007/s00014-003-0797-2
http://www.ems-ph.org/doi/10.1007/s00014-003-0797-2
Asymptotique des nombres de Betti, invariants $l^2$ et laminations
N.
Bergeron
Université Paris-Sud, ORSAY CX, FRANCE
Damien
Gaboriau
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
Betti numbers, finite covers, laminations, $l^2$-Betti numbers, measure preserving actions
Let K be a finite simplicial complex. We are interested in the asymptotic behavior of the Betti numbers of a sequence of finite sheeted covers of $K$, when normalized by the index of the covers. W. Lück, has proved that for regular coverings, these sequences of numbers converge to the $l^2$ Betti numbers of the associated (in general infinite) limit regular cover of K. In this article we investigate the non regular case. We show that the sequences of normalized Betti numbers still converge. But this time the good limit object is no longer the associated limit cover of K, but a lamination by simplicial complexes. We prove that the limits of sequences of normalized Betti numbers are equal to the $l^2$ Betti numbers of this lamination. Even if the associated limit cover of K is contractible, its $l^2$ Betti numbers are in general different from those of the lamination. We construct such examples. We also give a dynamical condition for these numbers to be equal. It turns out that this condition is equivalent to a former criterion due to M. Farber. We hope that our results clarify its meaning and show to which extent it is optimal. In a second part of this paper we study non free measure-preserving ergodic actions of a countable group $\Gamma$ on a standard Borel probability space. Extending group-theoretic similar results of the second author, we obtain relations between the $l^{2}$ Betti numbers of $\Gamma$ and those of the generic stabilizers. For example, if $b_1^{(2)} (\Gamma ) \neq 0$, then either almost each stabilizer is finite or almost each stabilizer has an infinite first $l^2$ Betti number.
Algebraic topology
Dynamical systems and ergodic theory
Manifolds and cell complexes
Global analysis, analysis on manifolds
362
395
10.1007/s00014-003-0798-1
http://www.ems-ph.org/doi/10.1007/s00014-003-0798-1
Varieties of pairs of nilpotent matrices annihilating each other
Jan
Schröer
University of Leeds, LEEDS, UNITED KINGDOM
Nilpotent matrix, irreducible component, Gelfand-Ponomarev algebra, string module, band module
We classify the irreducible components of the varieties \[ \V(n,a,b) = \{ (A,B) \in \M_n(\field) \times \M_n(\field) \mid AB = BA = A^a = B^b = 0 \}. \]
Algebraic geometry
Associative rings and algebras
General
396
426
10.1007/s00014-003-0788-3
http://www.ems-ph.org/doi/10.1007/s00014-003-0788-3
The representation theory of cyclotomic Temperley-Lieb algebras
HEBING
RUI
EAST CHINA NORMAL UNIVERSITY, SHANGHAI, CHINA
Changchang
Xi
Capital Normal University, BEIJING, CHINA
Temperley-Lieb algebra, cellular algebra, cyclotomic Temperley-Lieb algebra
A class of associative algebras called cyclotomic Temperley-Lieb algebras is introduced in terms of generators and relations. They are closely related to the group algebras of complex reflection groups on the one hand and generalizations of the usual Temperley-Lieb algebras on the other hand. It is shown that the cyclotomic Temperley-Lieb algebras can be defined by means of labelled Temperley-Lieb diagrams and are cellular in the sense of Graham and Lehrer. One thus obtains not only a description of the irreducible representations, but also a criterion for their quasi-heredity in the sense of Cline, Parshall and Scott. The branching rule for cell modules and the determinants of Gram matrices for certain cell modules are calculated, they can be expressed in terms of generalized Tchebychev polynomials, which therefore play an important role for semisimplicity.
Associative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
Group theory and generalizations
427
450
10.1007/s00014-004-0800-6
http://www.ems-ph.org/doi/10.1007/s00014-004-0800-6