Spaces of Graphs and Surfaces: on the Work of Søren Galatius
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Stable Homotopy of Complex Projective Space
THE STABLE HOMOTOPY OF COMPLEX PROJECTIVE SPACE By GRAEME SEGAL [Received 17 March 1972] 1. Introduction THE object of this note is to prove that the space BU is a direct factor of the space Q(CP°°) = Oto5eo(CP00) = HmQn/Sn(CPto). This is not very n surprising, as Toda [cf. (6) (2.1)] has shown that the homotopy groups of ^(CP00), i.e. the stable homotopy groups of CP00, split in the appro- priate way. But the method, which is Quillen's technique (7) of reducing to a problem about finite groups and then using the Brauer induction theorem, may be interesting. If X and Y are spaces, I shall write {X;7}*for where Y+ means Y together with a disjoint base-point, and [ ; ] means homotopy classes of maps with no conditions about base-points. For fixed Y, X i-> {X; Y}* is a representable cohomology theory. If Y is a topological abelian group the composition YxY ->Y induces and makes { ; Y}* into a multiplicative cohomology theory. In fact it is easy to see that {X; Y}° is then even a A-ring. Let P = CP00, and embed it in the space ZxBU which represents the functor K by P — IXBQCZXBU. This corresponds to the natural inclusion {line bundles} c {virtual vector bundles}. There is an induced map from the suspension-spectrum of P to the spectrum representing l£-theory, inducing a transformation of multiplicative cohomology theories T: { ; P}* -*• K*. PROPOSITION 1. For any space, X the ring-homomorphism T:{X;P}°-*K<>(X) is mirjective. COEOLLAKY. The space QP is (up to homotopy) the product of BU and a space with finite homotopy groups. -
4. Groups of Permutations 1
4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact. -
PERMUTATION GROUPS (16W5087)
PERMUTATION GROUPS (16w5087) Cheryl E. Praeger (University of Western Australia), Katrin Tent (University of Munster),¨ Donna Testerman (Ecole Polytechnique Fed´ erale´ de Lausanne), George Willis (University of Newcastle) November 13 - November 18, 2016 1 Overview of the Field Permutation groups are a mathematical approach to analysing structures by studying the rearrangements of the elements of the structure that preserve it. Finite permutation groups are primarily understood through combinatorial methods, while concepts from logic and topology come to the fore when studying infinite per- mutation groups. These two branches of permutation group theory are not completely independent however because techniques from algebra and geometry may be applied to both, and ideas transfer from one branch to the other. Our workshop brought together researchers on both finite and infinite permutation groups; the participants shared techniques and expertise and explored new avenues of study. The permutation groups act on discrete structures such as graphs or incidence geometries comprising points and lines, and many of the same intuitions apply whether the structure is finite or infinite. Matrix groups are also an important source of both finite and infinite permutation groups, and the constructions used to produce the geometries on which they act in each case are similar. Counting arguments are important when studying finite permutation groups but cannot be used to the same effect with infinite permutation groups. Progress comes instead by using techniques from model theory and descriptive set theory. Another approach to infinite permutation groups is through topology and approx- imation. In this approach, the infinite permutation group is locally the limit of finite permutation groups and results from the finite theory may be brought to bear. -
Binding Groups, Permutation Groups and Modules of Finite Morley Rank Alexandre Borovik, Adrien Deloro
Binding groups, permutation groups and modules of finite Morley rank Alexandre Borovik, Adrien Deloro To cite this version: Alexandre Borovik, Adrien Deloro. Binding groups, permutation groups and modules of finite Morley rank. 2019. hal-02281140 HAL Id: hal-02281140 https://hal.archives-ouvertes.fr/hal-02281140 Preprint submitted on 9 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BINDING GROUPS, PERMUTATION GROUPS AND MODULES OF FINITE MORLEY RANK ALEXANDRE BOROVIK AND ADRIEN DELORO If one has (or if many people have) spent decades classifying certain objects, one is apt to forget just why one started the project in the first place. Predictions of the death of group theory in 1980 were the pronouncements of just such amne- siacs. [68, p. 4] Contents 1. Introduction and background 2 1.1. The classification programme 2 1.2. Concrete groups of finite Morley rank 3 2. Binding groups and bases 4 2.1. Binding groups 4 2.2. Bases and parametrisation 5 2.3. Sharp bounds for bases 6 3. Permutation groups of finite Morley rank 7 3.1. Highly generically multiply transitive groups 7 3.2. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
6 Permutation Groups
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 6 Permutation Groups Let S be a nonempty set and M(S) be the collection of all mappings from S into S. In this section, we will emphasize on the collection of all invertible mappings from S into S. The elements of this set will be called permutations because of Theorem 2.4 and the next definition. Definition 6.1 Let S be a nonempty set. A one-to-one mapping from S onto S is called a permutation. Consider the collection of all permutations on S. Then this set is a group with respect to composition. Theorem 6.1 The set of all permutations of a nonempty set S is a group with respect to composition. This group is called the symmetric group on S and will be denoted by Sym(S). Proof. By Theorem 2.4, the set of all permutations on S is just the set I(S) of all invertible mappings from S to S. According to Theorem 4.3, this set is a group with respect to composition. Definition 6.2 A group of permutations , with composition as the operation, is called a permutation group on S. Example 6.1 1. Sym(S) is a permutation group. 2. The collection L of all invertible linear functions from R to R is a permu- tation group with respect to composition.(See Example 4.4.) Note that L is 1 a proper subset of Sym(R) since we can find a function in Sym(R) which is not in L, namely, the function f(x) = x3. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
CS E6204 Lecture 5 Automorphisms
CS E6204 Lecture 5 Automorphisms Abstract An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Under composition, the set of automorphisms of a graph forms what algbraists call a group. In this section, graphs are assumed to be simple. 1. The Automorphism Group 2. Graphs with Given Group 3. Groups of Graph Products 4. Transitivity 5. s-Regularity and s-Transitivity 6. Graphical Regular Representations 7. Primitivity 8. More Automorphisms of Infinite Graphs * This lecture is based on chapter [?] contributed by Mark E. Watkins of Syracuse University to the Handbook of Graph The- ory. 1 1 The Automorphism Group Given a graph X, a permutation α of V (X) is an automor- phism of X if for all u; v 2 V (X) fu; vg 2 E(X) , fα(u); α(v)g 2 E(X) The set of all automorphisms of a graph X, under the operation of composition of functions, forms a subgroup of the symmetric group on V (X) called the automorphism group of X, and it is denoted Aut(X). Figure 1: Example 2.2.3 from GTAIA. Notation The identity of any permutation group is denoted by ι. (JG) Thus, Watkins would write ι instead of λ0 in Figure 1. 2 Rigidity and Orbits A graph is asymmetric if the identity ι is its only automor- phism. Synonym: rigid. Figure 2: The smallest rigid graph. (JG) The orbit of a vertex v in a graph G is the set of all vertices α(v) such that α is an automorphism of G. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups. -
Lecture 9: Permutation Groups
Lecture 9: Permutation Groups Prof. Dr. Ali Bülent EKIN· Doç. Dr. Elif TAN Ankara University Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 1 / 14 Permutation Groups Definition A permutation of a nonempty set A is a function s : A A that is one-to-one and onto. In other words, a pemutation of a set! is a rearrangement of the elements of the set. Theorem Let A be a nonempty set and let SA be the collection of all permutations of A. Then (SA, ) is a group, where is the function composition operation. The identity element of (SA, ) is the identity permutation i : A A, i (a) = a. ! 1 The inverse element of s is the permutation s such that 1 1 ss (a) = s s (a) = i (a) . Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 2 / 14 Permutation Groups Definition The group (SA, ) is called a permutation group on A. We will focus on permutation groups on finite sets. Definition Let In = 1, 2, ... , n , n 1 and let Sn be the set of all permutations on f g ≥ In.The group (Sn, ) is called the symmetric group on In. Let s be a permutation on In. It is convenient to use the following two-row notation: 1 2 n s = ··· s (1) s (2) s (n) ··· Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 3 / 14 Symmetric Groups 1 2 3 4 1 2 3 4 Example: Let f = and g = . Then 1 3 4 2 4 3 2 1 1 2 3 4 1 2 3 4 1 2 3 4 f g = = 1 3 4 2 4 3 2 1 2 4 3 1 1 2 3 4 1 2 3 4 1 2 3 4 g f = = 4 3 2 1 1 3 4 2 4 2 1 3 which shows that f g = g f . -
Lectures on the Stable Homotopy of BG 1 Preliminaries
Geometry & Topology Monographs 11 (2007) 289–308 289 arXiv version: fonts, pagination and layout may vary from GTM published version Lectures on the stable homotopy of BG STEWART PRIDDY This paper is a survey of the stable homotopy theory of BG for G a finite group. It is based on a series of lectures given at the Summer School associated with the Topology Conference at the Vietnam National University, Hanoi, August 2004. 55P42; 55R35, 20C20 Let G be a finite group. Our goal is to study the stable homotopy of the classifying space BG completed at some prime p. For ease of notation, we shall always assume that any space in question has been p–completed. Our fundamental approach is to decompose the stable type of BG into its various summands. This is useful in addressing many questions in homotopy theory especially when the summands can be identified with simpler or at least better known spaces or spectra. It turns out that the summands of BG appear at various levels related to the subgroup lattice of G. Moreover since we are working at a prime, the modular representation theory of automorphism groups of p–subgroups of G plays a key role. These automorphisms arise from the normalizers of these subgroups exactly as they do in p–local group theory. The end result is that a complete stable decomposition of BG into indecomposable summands can be described (Theorem 6) and its stable homotopy type can be characterized algebraically (Theorem 7) in terms of simple modules of automorphism groups. This paper is a slightly expanded version of lectures given at the International School of the Hanoi Conference on Algebraic Topology, August 2004. -
The Rotation Group of Rubik's Cube
The Rotation Group of Rubik's Cube Oliver Knill Roman E. Mdder ETH Zurich Revised by Robin S. Sanders and Julie D. Simon University of Illinois, at Urbana-Champaign ABSTRACT In this project, the group theory program "Cayley" is used to solve some problems asso ciated with Rubik's Cube, including the original problem of restoring the initial state of the cube. Project-No.: 003-021 Title: Rubik's Cube Subject: Algebra Programs: Cayley References: 001-021 Die Drehgruppe des Rubikwurfels fypesetting: mltroff Version: 5/15/87 33 T h e M a t h e m a t i c a l L a b o r a t o r y R u b i k ' s C u b e 1. Introduction Several years have passed since the Rubik's Cube conquered the world. Almost everyone knows this cube and many have solved it or have tried to do so for hours. Rarely has there been a puzzle which has caused such a stir and which has found its way not only into the world of games but also into that of mathematics and computers. Before you start your work on this project about Rubik's cube, we recommend that you reach for a cube and get familiar with it (again). Many concepts of group theory can be demonstrated with this cube, and abstract definitions can become more intuitive. 1.1. The Cube If you already know it or are even a cube expert, you may skim this section which defines the structure of the cube and also introduces some notation.