DOCSLIB.ORG

Configuration Spaces in Algebraic Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Configuration Spaces in Algebraic Topology Harvard Math 252x Configuration Spaces in Algebraic Topology Taught by Ben Knudsen Notes by Reuben Stern September 4, 2017 Contents 0.1 Preliminaries ....................................... 1 1 Wednesday, August 301 1.1 A bunch of examples................................... 3 2 Friday, September 13 0.1 Preliminaries This is Math 252x, configuration spaces in algebraic topology, taught by Ben Knudsen. The goal is to understand the homotopy and cohomology of configuration spaces. Office hours will be on Thursdays from 1-3 pm. 1 Wednesday, August 30 Definition 1.1. The configuration space of k ordered points in the topological space X is k Confk(X) = {(x1, . , xn) ∈ X : xi 6= xj if i 6= j}. The unordered configuration space is Bk(X) := Confk(X)/Σk. m ∼ n Aside 1.2. As we’ll see, B2(R ) = B2(R ) if and only if n = m, so configuration spaces are 1 sensitive to the dimension of manifolds. A homology computation shows 2 2 0 B2(T \ {∗}) 6' B2(R \ S ), even though the two have the same dimension and share S1 ∨ S1 as a common deformation retract. Example 1.3. The (p, q)-lens space is the quotient 3 L(p, q) := S /Cp, 3 2 2πi/p 2πiq/p where Cp acts on S ⊆ C by multiplication by (e , e ). 2 Theorem 1.4 (Reidemeister). L(p, q1) ' L(p, q2) if and only if q1q2 ≡ ±n mod p, and ∼ ±1 L(p, q1) = L(p, q2) if and only if q1 ≡ ±q2 mod p. For instance, L(7, 1) and L(7, 2) are homotopy equivalent but not homeomorphic. Theorem 1.5 (Longoni-Salvatore). B2(L(7, 1)) 6' B2(L(7, 2)). So there are two compact manifolds that have the same homotopy type but different configura- tion spaces. Question 1.6. How much do configuration spaces know of the background manifold? Perspective 1.7. Let’s consider braids: they can wind around each other, but cannot intersect. 2 It follows that π1(Bk(R )) is the braid group on k strands. Artin showed that 2 ∼ π1(Bk(R )) = hσ1, . , σk−1|σiσj = σjσi if |i − j| > 1; σiσj + ...i Braid-like groups are fundamentally important in knot theory, mapping class groups, quantum groups, category theory (braided monoidal categories), and motion planning and robotics. Perspective 1.8. (Embeddings). There is a space of framed embeddings of a manifold with a fr ` n n n fixed trivialization of the tangent bundle, i.e., Emb ( k R , R ), which maps to Confk(R ), and in fact is a homotopy equivalence. Since embeddings can be composed, the collection n {Confk(R )}k≥0 of homotopy types has a hidden algebraic structure of an En-operad. There are a lot of things to say now: 1. Iterated loop spaces a la May. It turns out that ! a Embfr Rn, Rn × (ΩnX)k −→ ΩnX k n tells us that Ω X is an En-algebra. This has two consequences: n n • H∗(Confk(R )) provides operations in H∗(Ω X). • From the recognition theorem, there is an algebraic classification of n-fold loop spaces. 2 2. Factorization homology (Lurie, Ayala-Francis), also known as topological chiral homology. R R Suppose I have an En-algebra A; this gives a manifold invariant M A. Roughly, M A is a space of configurations of points in M, labeled by A. R n • M Ω X ' Mapc(M, X) if X is (n − 1)-connected (called non-abelian Poincaré duality, due to Lurie and Salvatore). • Applications to configuration spaces. 3. Embedding calculus (Goodwillie, Weiss). Suppose you want to understand Emb(M, N). We get a tower of approximations . T2Emb(M, N) Emb(M, N) T1Emb(M, N). 1.1 A bunch of examples Example 1.9. Take a look at the empty manifold. Bk(?) = Confk(?) ' {∗} if k = 0, and it is 0 empty otherwise. This follows from the convention ? = {∗}. Example 1.10. Now look at R0. 0 {∗} k = 0, 1 Confk(R ) = ? k > 1. Example 1.11. R1. The configuration space of 2 points is just what you think it would look like: 1 ∼ 2 the disjoint union of the interiors of two 2-simplices. And B2(R ) = ∆ is contractible. Now what if k > 2? We have a map π0Confk((0, 1)) → Σk, which is actually a bijection. Each component is homeomorphic to Bk((0, 1)). There is a map k Bk((0, 1)) −→ ∆ taking (x1, . , xk) to (x1, x2 − x1,..., 1 − xk). This is a homeomorphism. 1 1 2 1 Example 1.12. Now consider S . Conf2(S ) = T \ S , which is a cylinder, so is homotopy 1 1 equivalent to S . Similarly, the unordered configuration space B2(S ) is the mobius band. n n n−1 n Example 1.13. Now look at R . Conf2(R ) maps to S ×R>0×R , which is a homeomorphism. So the (x , x ) 7→ x2−x1 is a homotopy equivalence. Gauss map 1 2 kx2−x1k 2 Friday, September 1 Recall the definition of a configuration space from last time: 3 Definition 2.1. Let X be a topological space. The (ordered) configuration space of k k points in X is the set Confk(X) = {(x1, . , xk) ∈ X : xi = xj iff i = j}. The unordered configuration space Bk(X) is the quotient of Confk(X) by the natural action of the symmetric group Σk. One thing we can ask is about functoriality of this construction: is Confk(X) functorial? It is to a certain extent: if f : X → Y is an injective map of topological spaces, then there is a map Confk(f) filling in the dashed arrow in the commutative diagram f k Xk Y k ⊆ ⊆ Confk(X) Confk(Y ), Confk(f) given by applying f coordinate-wise. It is clear that this induced map is Σk-equivariant, so it descends to a map Bk(f): Bk(X) → Bk(Y ). Thus we have two functors: inj Σk Confk(−):Top −→ Top inj Bk(−):Top −→ Top. Here, Topinj is the category of topological spaces and injective maps, and TopΣk is the category of topological spaces with a Σk-action, and Σk-equivariant maps. Proposition 2.2. If f : X → Y is an open embedding, then so are Confk(f) and Bk(f). k k Proof. For Confk(f), note that f is an open embedding, and Confk(X) is open in X . For Bk(f), the natural map Confk(X) → Bk(X) is a quotient map, and thus an open embedding. Open embeddings compose. Definition 2.3. An injective, continuous map f : X → Y is called an isotopy equivalence if there is an injective, continuous map g : Y → X and homotopies 0 H : f ◦ g ⇒ 1Y and H : g ◦ f ⇒ 1X 0 such that Ht and Ht are injective for all t ∈ [0, 1]. Basically, it is a “homotopy equivalence through embeddings”. Proposition 2.4. If f : X → Y is an isotopy equivalence, the induced maps Confk(f) and Bk(f) on configuration spaces are homotopy equivalences. Proof. Consider the diagram (H0)k (X × I)k Xk H∆ Xk × Ik i i×∆ Confk(X) × I Confk(X) 4 ∆ k ∆ 0 0 The composite map H : Confk(X) × I → X acts by H (x1, . , xk, t) = (Ht(x1),...,Ht(xk)). 0 k Because Ht is injective, this map pulls back along the inclusion i : Confk(X) → X to give a well-defined homotopy Confk(X) × I → Confk(X). Example 2.5. Let M be a manifold with nonempty boundary. Then ∂M has a collar neighborhood U =∼ ∂M × (0, 1]. By retracting along the (0, 1] coordinate, the inclusion M,˚ → M is an isotopy equivalence, so ' Confk(M˚) Confk(M). Notation 2.6. Fix a basis B of the topology of X. Define two partially-ordered sets Bk and Σ Bk as follows: ( open k ) ∼ G 1. Bk = Uk ⊆ X : U = Ui,Ui ∈ B . We order this by saying U ≤ V if U ⊆ V and the i=1 inclusion i : U,→ V induces a bijection on π0. Σ n ' o 2. Bk = (U, σ): U ∈ Bk, σ : {1, . , k} −−→ π0(U) . This is ordered by saying (U, σ) ≤ (V, τ) if U ≤ V in Bk and i ◦ σ = τ, where i : U → V is the inclusion. Now let O(X) denote the poset of open subsets of X. We will define two functions, ◦ Bk :Bk ,−→O(Bk(X)) and ◦ Σ Confk :Bk ,−→O(Confk(X)) as follows: ◦ Bk(U) = {x ∈ Bk(U): x intersects each component of U nontrivially} ◦ n o Confk(U, σ) = (x1, . , xk) ∈ Confk(U): xi ∈ Uσ(i) . k ◦ ∼ Y ∼ ◦ Lemma 2.7. There are homeomorphisms Confk(U, σ) = Uσ(i) = Bk(U). i=1 Proof. The proof starts by considering the diagram k X Confk(X) Bk(X) k Y ◦ ◦ Uσ(i) Confk(U, σ) Bk(U). i=1 The dashed arrows can easily be seen to be filled in. finish this proof Σ Proposition 2.8. The set Bk is a basis for the topology of Confk(X). The set Bk is a basis for the topology of Bk. Proof. Finish later. 5 Corollary 2.9. The projection π : Confk(X) → Bk(X) is a covering space map. Proof. −1 ◦ ∼ G ◦ π (Bk(U)) = Confk(U, σ) σ ∼ ◦ = Bk(U) × Σk. Corollary 2.10. k G =∼ Confi(X) × Confk−i(Y ) ×Σi×Σk−i ΣK Confk(X t Y ). i=0 In particular, we have a homeomorphism k G =∼ Bi(X) × Bk−i(Y ) Bk(X t Y ). i=0 Now, let ` ≥ k be integers. The projection map X` → Xk onto the first k coordinates induces a dashed map on configuration spaces: ` Conf`(X) X (x1, .
Load more

Recommended publications
  • Sheaves and Homotopy Theory
    SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case.
    [Show full text]
  • 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
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology
    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
    [Show full text]
  • 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 .
    [Show full text]
  • Topology - Wikipedia, the Free Encyclopedia Page 1 of 7
    Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds.
    [Show full text]
  • ANALYTIC TORSION 0. Introduction Analytic Torsion Is an Invariant Of
    ANALYTIC TORSION SAIF SULTAN Abstract. This article is a survey of analytic torsion of elliptic operator com- plexes. The scope of the article is expository. We define the analytic torsion of elliptic complexes in general and with this define the analytic torsion in terms of the spectrum of the Laplace operator on a Riemannian Manifold. We define the Riedemeister Torsion of a Manifold and discuss the celebrated Cheeger-Mueller theorem relating analytic torsion and Riedemeister torsion. 0. Introduction Analytic Torsion is an invariant of Riemannian Manifolds, first introduced by Ray and Singer in 1970s. It is defined in terms of determinants of Laplacians on n-forms of a Riemannian Manifold M. Using same construction it is defined for general elliptic complexes. Analytic torsion was introduced as the analytic version of the Reidemeister torsion (R-torsion). R-torsion is an algebraic topology invariant introduced by Reidemeister in 1935 [1] in classification of three dimensional lens spaces using simplicial chain complex of universal covers. Lens spaces were first known examples of 3-manifolds whose homotopy type and homology do not determine their homeomorphism type. R- torsion was generalized by Franz [2] in same year, to higher dimensional lens spaces. There is a complete classification of 3dimensional lens spaces in terms of R-torsion and fundamental group. There exist homotopic lens space with different R-torions. Though R-torsion is not a homotpy invariant, it was shown to be a topological invariant by Kirby and Siebenmann in 1969. In 1971 D.B. Ray and I.M. Singer conjectured the equality of the two torsions, which was proved independently J.
    [Show full text]
  • Homotopy Theory of Infinite Dimensional Manifolds?
    Topo/ogy Vol. 5. pp. I-16. Pergamon Press, 1966. Printed in Great Britain HOMOTOPY THEORY OF INFINITE DIMENSIONAL MANIFOLDS? RICHARD S. PALAIS (Received 23 Augrlsf 1965) IN THE PAST several years there has been considerable interest in the theory of infinite dimensional differentiable manifolds. While most of the developments have quite properly stressed the differentiable structure, it is nevertheless true that the results and techniques are in large part homotopy theoretic in nature. By and large homotopy theoretic results have been brought in on an ad hoc basis in the proper degree of generality appropriate for the application immediately at hand. The result has been a number of overlapping lemmas of greater or lesser generality scattered through the published and unpublished literature. The present paper grew out of the author’s belief that it would serve a useful purpose to collect some of these results and prove them in as general a setting as is presently possible. $1. DEFINITIONS AND STATEMENT OF RESULTS Let V be a locally convex real topological vector space (abbreviated LCTVS). If V is metrizable we shall say it is an MLCTVS and if it admits a complete metric then we shall say that it is a CMLCTVS. A half-space in V is a subset of the form {v E V(l(u) L 0} where 1 is a continuous linear functional on V. A chart for a topological space X is a map cp : 0 + V where 0 is open in X, V is a LCTVS, and cp maps 0 homeomorphically onto either an open set of V or an open set of a half space of V.
    [Show full text]
  • 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.
    [Show full text]
  • Local Homotopy Theory, Iwasawa Theory and Algebraic K–Theory
    K(1){local homotopy theory, Iwasawa theory and algebraic K{theory Stephen A. Mitchell? University of Washington, Seattle, WA 98195, USA 1 Introduction The Iwasawa algebra Λ is a power series ring Z`[[T ]], ` a fixed prime. It arises in number theory as the pro-group ring of a certain Galois group, and in homotopy theory as a ring of operations in `-adic complex K{theory. Fur- thermore, these two incarnations of Λ are connected in an interesting way by algebraic K{theory. The main goal of this paper is to explore this connection, concentrating on the ideas and omitting most proofs. Let F be a number field. Fix a prime ` and let F denote the `-adic cyclotomic tower { that is, the extension field formed by1 adjoining the `n-th roots of unity for all n 1. The central strategy of Iwasawa theory is to study number-theoretic invariants≥ associated to F by analyzing how these invariants change as one moves up the cyclotomic tower. Number theorists would in fact consider more general towers, but we will be concerned exclusively with the cyclotomic case. This case can be viewed as analogous to the following geometric picture: Let X be a curve over a finite field F, and form a tower of curves over X by extending scalars to the algebraic closure of F, or perhaps just to the `-adic cyclotomic tower of F. This analogy was first considered by Iwasawa, and has been the source of many fruitful conjectures. As an example of a number-theoretic invariant, consider the norm inverse limit A of the `-torsion part of the class groups in the tower.
    [Show full text]
  • 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.
    [Show full text]