ANALYTIC TORSION 0. Introduction Analytic Torsion Is an Invariant Of
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Notices of the American Mathematical Society
OF THE AMERICAN MATHEMATICAL SOCIETY ISSU! NO. 116 OF THE AMERICAN MATHEMATICAL SOCIETY Edited by Everett Pitcher and Gordon L. Walker CONTENTS MEETINGS Calendar of Meetings ••••••••••••••••••••••••••••••••••.• 874 Program of the Meeting in Cambridge, Massachusetts •••.•.••••..•• 875 Abstracts for the Meeting- Pages 947-953 PRELIMINARY ANNOUNCEMENTS OF MEETINGS •••••••••••••••••.•• 878 AN APPEAL FOR PRESERVATION OF ARCHIVAL MATERIALS .•••••••••• 888 CAN MATHEMATICS BE SAVED? ••••••••••.••••••••..•.•••••••.. 89 0 DOCTORATES CONFERRED IN 1968-1969 ••••••••••••••.••••••.•••• 895 VISITING MATHEMATICIANS .•••••••••••••••••••••••••..•••••.. 925 ANNUAL SALARY SURVEY ••••••••••••.••••.••••.•.•.••••••.•• 933 PERSONAL ITEMS •••••••••••••••••••••••••••••...•••••••••• 936 MEMORANDA TO MEMBERS Audio Recordings of Mathematical Lectures ••••••••..•••••.•••.• 940 Travel Grants. International Congress of Mathematicians ••..•.•••••.• 940 Symposia Information Center ••••.•• o o • o ••••• o o •••• 0 •••••••• 940 Colloquium Lectures •••••••••••••••••••••••.• 0 ••••••••••• 941 Mathematical Sciences E'mployment Register .•.••••••..•. o • o ••••• 941 Retired Mathematicians ••••• 0 •••••••• 0 ••••••••••••••••• 0 •• 942 MOS Reprints .•••••• o •• o ••••••••••••••••••••••• o •••••• 942 NEWS ITEMS AND ANNOUNCEMENTS •••••. o •••••••••••••••• 877, 932, 943 ABSTRACTS PRESENTED TO THE SOCIETY •••••.••••.•.•.••..•..•• 947 RESERVATION FORM. o •••••••••••••••••••••••••••••••••••••• 1000 MEETINGS Calendar of Meetings NOTE: This Calendar lists all of the meetings which have -
A Conjecture on the Equivariant Analytic Torsion Forms
A conjecture on the equivariant analytic torsion forms Vincent Maillot & Damian R¨ossler ∗ September 21, 2005 Let G be a finite group and let M be a complex manifold on which G acts by holomorphic automorphisms. Let f : M → B be a proper holomorphic map of complex manifolds and suppose that G preserves the fibers of f (i.e. f ◦ g = f for all g ∈ G). Let η be a G-equivariant holomorphic vector bundle on M and • η suppose that the direct images R f∗η are locally free on B. Let h be a G- invariant hermitian metric on η and let ωM be a G-invariant K¨ahlermetric on M. For each g ∈ G, we shall write Mg for the set of fixed points of g on M. This set is endowed with a natural structure of complex K¨ahlermanifold. We suppose that there is an open dense set U of B such that the restricted map −1 −1 f (U) → U is a submersion. We shall write V for f (U) and fV for the map −1 V η U fV : f (U) → U. Denote by Tg(ω , h ) ∈ P the equivariant analytic torsion V M form of η|V relatively to fV and ω := ω |V , in the sense of [4, Par. d)]. Here the space P U (resp. P U,0) is the direct sum of the space of complex differential forms of type p, p (resp. the direct sum of the space of complex differential forms V η of type p, p of the form ∂α + ∂β) on U. -
DEGREE-L MAPS INTO LENS SPACES and FREE CYCLIC ACTIONS on HOMOLOGY 3-SPHERES*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 37 (1990) 131-136 131 North-Holland DEGREE-l MAPS INTO LENS SPACES AND FREE CYCLIC ACTIONS ON HOMOLOGY 3-SPHERES* E. LUST and D. SJERVE Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada V6T 1 Y4 Received 20 September 1988 Revised 7 August 1989 If M is a closed orientable 3-manifold with H,(M) = Z,, then there is a lens space L,, unique up to homotopy, and a degree-l map f: A4+ L,,. As a corollary we prove that if 6 is a homology 3-sphere admitting a fixed point free action by Z,, then the regular covering fi + It?/??, is induced from the universal covering S3 --f L,,,, by a degree-l map f: 6/Z, + L,,,. AMS (MOS) Subj. Class.: Primary 57M99 homology 3-spheres degree-l maps lens spaces 1. Introduction In this paper we will be concerned with the existence of degree-l mapsf: M + L,, where M is a closed orientable 3-manifold and L,, is some lens space. If H,(M) = Z,, such maps exist and the lens space L,, is unique up to homotopy equivalence (see Theorem 3.4). If fi is a homology 3-sphere and p : A? -+ M is a regular covering with the cyclic group Z,, as its covering group, then H,(M) =Z,, and the covering p : A? + M is induced from the universal covering q : S3 -+ L,, by means of a degree-l map f: M + L,, (see Th eorem 3.5). -
LEGENDRIAN LENS SPACE SURGERIES 3 Where the Ai ≥ 2 Are the Terms in the Negative Continued Fraction Expansion P 1 = A0 − =: [A0,...,Ak]
LEGENDRIAN LENS SPACE SURGERIES HANSJORG¨ GEIGES AND SINEM ONARAN Abstract. We show that every tight contact structure on any of the lens spaces L(ns2 − s + 1,s2) with n ≥ 2, s ≥ 1, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot T (s, −(sn − 1)) in the tight or an overtwisted contact structure on the 3-sphere. 1. Introduction A knot K in the 3-sphere S3 is said to admit a lens space surgery if, for some rational number r, the 3-manifold obtained by Dehn surgery along K with surgery coefficient r is a lens space. In [17] L. Moser showed that all torus knots admit lens space surgeries. More precisely, −(ab ± 1)-surgery along the negative torus knot T (a, −b) results in the lens space L(ab ± 1,a2), cf. [21]; for positive torus knots one takes the mirror of the knot and the surgery coefficient of opposite sign, resulting in a negatively oriented lens space. Contrary to what was conjectured by Moser, there are surgeries along other knots that produce lens spaces. The first example was due to J. Bailey and D. Rolfsen [1], who constructed the lens space L(23, 7) by integral surgery along an iterated cable knot. The question which knots admit lens space surgeries is still open and the subject of much current research. The fundamental result in this area is due to Culler– Gordon–Luecke–Shalen [2], proved as a corollary of their cyclic surgery theorem: if K is not a torus knot, then at most two surgery coefficients, which must be successive integers, can correspond to a lens space surgery. -
Knots and Links in Lens Spaces
Alma Mater Studiorum Università di Bologna Dottorato di Ricerca in MATEMATICA Ciclo XXVI Settore Concorsuale di afferenza: 01/A2 Settore Scientifico disciplinare: MAT/03 Knots and links in lens spaces Tesi di Dottorato presentata da: Enrico Manfredi Coordinatore Dottorato: Relatore: Prof.ssa Prof. Giovanna Citti Michele Mulazzani Esame Finale anno 2014 Contents Introduction 1 1 Representation of lens spaces 9 1.1 Basic definitions . 10 1.2 A lens model for lens spaces . 11 1.3 Quotient of S3 model . 12 1.4 Genus one Heegaard splitting model . 14 1.5 Dehn surgery model . 15 1.6 Results about lens spaces . 17 2 Links in lens spaces 19 2.1 General definitions . 19 2.2 Mixed link diagrams . 22 2.3 Band diagrams . 23 2.4 Grid diagrams . 25 3 Disk diagram and Reidemeister-type moves 29 3.1 Disk diagram . 30 3.2 Generalized Reidemeister moves . 32 3.3 Standard form of the disk diagram . 36 3.4 Connection with band diagram . 38 3.5 Connection with grid diagram . 42 4 Group of links in lens spaces via Wirtinger presentation 47 4.1 Group of the link . 48 i ii CONTENTS 4.2 First homology group . 52 4.3 Relevant examples . 54 5 Twisted Alexander polynomials for links in lens spaces 57 5.1 The computation of the twisted Alexander polynomials . 57 5.2 Properties of the twisted Alexander polynomials . 59 5.3 Connection with Reidemeister torsion . 61 6 Lifting links from lens spaces to the 3-sphere 65 6.1 Diagram for the lift via disk diagrams . 66 6.2 Diagram for the lift via band and grid diagrams . -
L2-Analytic Torsion
JOURNAL OF FUNCTIONAL ANALYSIS 107, 369-386 (1992) L2-Analytic Torsion VARGHESE MATHAI Department of Pure Mathematics, University of Adelaide. Adelaide, South Australia, Australia Communicated by R. B. Melrose Received May 8, 1991 In this paper, we introduce a new differential invariant called L*-analytic torsion, for closed manifolds with positive decay and whose universal covers have trivial L*-cohomology. L*-analytic torsion can be thought of as a suitable generalization of the Ray-Singer analytic torsion. We establish various functorial properties of L*-analytic torsion and also compute it for odd-dimensional, closed, hyperbolic manifolds, with the help of results from Fried. d-’ 1992 Academic Press, Inc. INTRODUCTION In [3], we studied a class of closed, oriented manifolds, called L2-acyclic manifolds, which have the characterising property that there are no harmonic L2-differential forms on the universal cover. It follows from a theorem of Dodzuik that the condition of L2-acyclicity depends only on the homotopy type of the closed manifold. In this paper we study a differential invariant of a class of manifolds which are L2-acyclic and which have positive decay. The condition of positive decay states that the heat kernel of the Laplacian on forms (on the universal cover) decays faster than some negative power of time to the projection to the L2-harmonic differential forms on the universal cover, uniformly over compact subsets of the universal cover. By a theorem of Novikov and Shubin, it follows that the condition of positive decay is inde- pendent of the choice of metric on the manifold, and hence it only depends on the smooth structure. -
Analytic Torsion and Dynamical Flow: a Survey on the Fried Conjecture
ANALYTIC TORSION AND DYNAMICAL FLOW: A SURVEY ON THE FRIED CONJECTURE SHU SHEN ABSTRACT. Given an acyclic and unitarily flat vector bundle on a closed manifold, Fried conjectured an equality between the analytic torsion and the value at zero of the Ruelle zeta function associated to a dynamical flow. In this survey, we review the Fried conjecture for different flows, including the suspension flow, the Morse-Smale flow, the geodesic flow, and the Anosov flow. CONTENTS Introduction2 1. The Ray-Singer analytic torsion5 1.1. The flat vector bundle5 1.2. Hodge Laplacian and analytic torsion5 1.3. A formula for fibrations7 2. The Fried conjecture8 2.1. Lefschetz index8 2.2. Fuller index9 2.3. The Ruelle dynamical zeta function 11 3. Suspension flow of a diffeomorphism 12 3.1. The definition of the suspension flow 12 3.2. The flat vector bundle on the suspension 13 3.3. The Ruelle dynamical zeta function of the suspension flow 14 4. Morse-Smale flow 15 4.1. The definition of Morse-Smale flow 16 4.2. The Ruelle dynamical zeta function of the Morse-Smale flow 17 5. Geodesic flow 18 5.1. Flat vector bundle on the unit tangent bundle 19 5.2. Geodesic flow of the non positively curved manifolds 21 5.3. The V -invariant and a formal proof of the Fried conjecture 22 5.4. Reductive group, globally and locally symmetric space 23 5.5. The Selberg trace formula 26 5.6. Moscovici-Stanton’s vanishing theorem 30 5.7. The case G = SO0(p; 1) with p odd 31 5.8. -
Heegaard Splittings of Branched Cyclic Coverings of Connected Sums of Lens Spaces
Bull. Korean Math. Soc. 54 (2017), No. 5, pp. 1851{1857 https://doi.org/10.4134/BKMS.b160760 pISSN: 1015-8634 / eISSN: 2234-3016 HEEGAARD SPLITTINGS OF BRANCHED CYCLIC COVERINGS OF CONNECTED SUMS OF LENS SPACES Tatyana Kozlovskaya Abstract. We study relations between two descriptions of closed ori- entable 3-manifolds: as branched coverings and as Heegaard splittings. An explicit relation is presented for a class of 3-manifolds which are branched cyclic coverings of connected sums of lens spaces, where the branching set is an axis of a hyperelliptic involution of a Heegaard sur- face. 1. Introduction Arbitrary closed orientable 3-manifold M can be described in various ways: by its triangulation, fundamental polyhedron, surgery on a link, Heegaard split- ting, etc. Choosing of a way depends on a context as well as on a question asked about M. Closed orientable 3-manifolds with cyclic symmetries are objects of intensive study in last decades. The initial and most known examples of closed orientable 3-manifolds belong to the class of branched cyclic covers of the 3-sphere S3. Among them are the following: • spherical and hyperbolic dodecahedral spaces, constructed by Weber and Seifert in 1933 [14], are the 3-fold cyclic cover of S3 branched over the trefoil knot and the 5-fold cyclic cover of S3 branched over the Whitehead link, respectively; • Fibonacci manifolds, constructed by Helling, Kim, and Mennicke [8], are n-fold cyclic covers of S3 branched over the figure-eight knot; • Sieradski manifolds, constructed by Cavicchioli, Kim, and Hegenbarth [7], are n-fold cyclic covers of S3 branched over the trefoil knot; • the smallest volume closed orientable hyperbolic 3-manifold, construc- ted by Fomenko and Matveev [10] and by Weeks [15], is the 3-fold cyclic cover of S3, branched over the two-bridge knot 7=3. -
Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics
Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics by Phillip Andreae Department of Mathematics Duke University Date: Approved: Mark Stern, Supervisor Hubert Bray Lenhard Ng Leslie Saper Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2016 Abstract Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics by Phillip Andreae Department of Mathematics Duke University Date: Approved: Mark Stern, Supervisor Hubert Bray Lenhard Ng Leslie Saper An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2016 Copyright c 2016 by Phillip Andreae All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense. iv To my mother and father v Contents Abstract iv List of Symbols ix Acknowledgementsx 1 Introduction1 2 Vector bundles, metrics, and differential forms on spaces of metrics6 2.1 Differential forms.............................6 2.2 Sobolev spaces and algebras of operators................9 2.3 Adjoints and traces........................... -
A Legendrian Turaev Torsion Via Generating Families Tome 8 (2021), P
Daniel Álvarez-Gavela & Kiyoshi Igusa A Legendrian Turaev torsion via generating families Tome 8 (2021), p. 57-119. <http://jep.centre-mersenne.org/item/JEP_2021__8__57_0> © Les auteurs, 2021. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D’ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Tome 8, 2021, p. 57–119 DOI: 10.5802/jep.141 A LEGENDRIAN TURAEV TORSION VIA GENERATING FAMILIES by Daniel Álvarez-Gavela & Kiyoshi Igusa To our teachers Yasha Eliashberg and Allen Hatcher Abstract. — We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2- dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain mon- odromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1-jet space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian in- variant, yet are not Legendrian isotopic. -
Mathematics of the Gateway Arch Page 220
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice. -
Heat Kernels on Covering Spaces and Topological Invariants
J. DIFFERENTIAL GEOMETRY 35(1992) 471-510 HEAT KERNELS ON COVERING SPACES AND TOPOLOGICAL INVARIANTS JOHN LOTT I. Introduction It is well known that there are relationships between the heat flow, acting on differential forms on a closed oriented manifold M, and the topology of M. From Hodge theory, one can recover the Betti numbers of M from the heat flow. Furthermore, Ray and Singer [42] defined an analytic tor- sion, a smooth invariant of acyclic flat bundles on M, and conjectured that it equals the classical Reidemeister torsion. This conjecture was proved to be true independently by Cheeger [7] and Muller [40]. The analytic torsion is nonzero only on odd-dimensional manifolds, and behaves in some ways as an odd-dimensional counterpart of the Euler characteristic [20]. If M is not simply-connected, then there is a covering space analog of the Betti numbers. Using the heat flow on the universal cover M, one can define the L2-Betti numbers of M [1] by taking the trace of the heat kernel not in the ordinary sense, but as an element of a certain type II von Neumann algebra. More concretely, this amounts to integrating the local trace of the heat kernel over a fundamental domain in M. In §11, we summarize this theory. We consider the covering space analog of the analytic torsion. This iΛanalytic torsion has the same relation to the iΛcohomology as the ordinary analytic torsion bears to de Rham cohomology. In §111 we define the iΛanalytic torsion <9^(M), under a technical assumption which we discuss later, and prove its basic properties.