Lectures on the Stable Homotopy of BG 1 Preliminaries
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The Fundamental Group
The Fundamental Group Tyrone Cutler July 9, 2020 Contents 1 Where do Homotopy Groups Come From? 1 2 The Fundamental Group 3 3 Methods of Computation 8 3.1 Covering Spaces . 8 3.2 The Seifert-van Kampen Theorem . 10 1 Where do Homotopy Groups Come From? 0 Working in the based category T op∗, a `point' of a space X is a map S ! X. Unfortunately, 0 the set T op∗(S ;X) of points of X determines no topological information about the space. The same is true in the homotopy category. The set of `points' of X in this case is the set 0 π0X = [S ;X] = [∗;X]0 (1.1) of its path components. As expected, this pointed set is a very coarse invariant of the pointed homotopy type of X. How might we squeeze out some more useful information from it? 0 One approach is to back up a step and return to the set T op∗(S ;X) before quotienting out the homotopy relation. As we saw in the first lecture, there is extra information in this set in the form of track homotopies which is discarded upon passage to [S0;X]. Recall our slogan: it matters not only that a map is null homotopic, but also the manner in which it becomes so. So, taking a cue from algebraic geometry, let us try to understand the automorphism group of the zero map S0 ! ∗ ! X with regards to this extra structure. If we vary the basepoint of X across all its points, maybe it could be possible to detect information not visible on the level of π0. -
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 Homotopy Theory Primer
4 Homotopy theory primer Given that some topological invariant is different for topological spaces X and Y one can definitely say that the spaces are not homeomorphic. The more invariants one has at his/her disposal the more detailed testing of equivalence of X and Y one can perform. The homotopy theory constructs infinitely many topological invariants to characterize a given topological space. The main idea is the following. Instead of directly comparing struc- tures of X and Y one takes a “test manifold” M and considers the spacings of its mappings into X and Y , i.e., spaces C(M, X) and C(M, Y ). Studying homotopy classes of those mappings (see below) one can effectively compare the spaces of mappings and consequently topological spaces X and Y . It is very convenient to take as “test manifold” M spheres Sn. It turns out that in this case one can endow the spaces of mappings (more precisely of homotopy classes of those mappings) with group structure. The obtained groups are called homotopy groups of corresponding topological spaces and present us with very useful topological invariants characterizing those spaces. In physics homotopy groups are mostly used not to classify topological spaces but spaces of mappings themselves (i.e., spaces of field configura- tions). 4.1 Homotopy Definition Let I = [0, 1] is a unit closed interval of R and f : X Y , → g : X Y are two continuous maps of topological space X to topological → space Y . We say that these maps are homotopic and denote f g if there ∼ exists a continuous map F : X I Y such that F (x, 0) = f(x) and × → F (x, 1) = g(x). -
Some Finiteness Results for Groups of Automorphisms of Manifolds 3
SOME FINITENESS RESULTS FOR GROUPS OF AUTOMORPHISMS OF MANIFOLDS ALEXANDER KUPERS Abstract. We prove that in dimension =6 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of Rn and various types of automorphisms of 2-connected manifolds. Contents 1. Introduction 1 2. Homologically and homotopically finite type spaces 4 3. Self-embeddings 11 4. The Weiss fiber sequence 19 5. Proofs of main results 29 References 38 1. Introduction Inspired by work of Weiss on Pontryagin classes of topological manifolds [Wei15], we use several recent advances in the study of high-dimensional manifolds to prove a structural result about diffeomorphism groups. We prove the classifying spaces of such groups are often “small” in one of the following two algebro-topological senses: Definition 1.1. Let X be a path-connected space. arXiv:1612.09475v3 [math.AT] 1 Sep 2019 · X is said to be of homologically finite type if for all Z[π1(X)]-modules M that are finitely generated as abelian groups, H∗(X; M) is finitely generated in each degree. · X is said to be of finite type if π1(X) is finite and πi(X) is finitely generated for i ≥ 2. Being of finite type implies being of homologically finite type, see Lemma 2.15. Date: September 4, 2019. Alexander Kupers was partially supported by a William R. -
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. -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
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]. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
Properties of 3-Manifolds for Relativists
Freiburg, THEP- 93/15 gr-qc/9308008 PROPERTIES OF 3-MANIFOLDS FOR RELATIVISTS Domenico Giulini* Fakult¨at f¨ur Physik, Universit¨at Freiburg Hermann-Herder Strasse 3, D-79104 Freiburg, Germany Abstract In canonical quantum gravity certain topological properties of 3-manifolds are of interest. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3-manifolds. We give a summary of these properties and list some old and new results concerning them. The appendix contains a discussion of the group of large diffeomorphisms of the l-handle 3- manifold. Introduction In the canonical formulation of general relativity and in particular in all approaches to canonical quantum gravity it emerges that the diffeomorphism groups of closed 3- manifolds are of particular interest. Here one is interested in a variety of questions concerning either the whole diffeomorphism group or certain subgroups thereof. More arXiv:gr-qc/9308008v1 9 Aug 1993 precisely, one is e.g. interested in whether the 3-manifold admits orientation reversing self-diffeomorphisms [So] (following this reference we will call a manifold chiral, iff it admits no such diffeomorphism) or what diffeomorphisms not connected to the identity there are in the subgroup fixing a frame [FS][Is][So][Wi1]. Other topological invariants of this latter subgroup are also argued to be of interest in quantum gravity [Gi1]. In the sum over histories approach one is also interested in whether a given 3-manifold can be considered as the spatial boundary of a spin-Lorentz 4-manifold [GH][Gi2] (following ref. -
The Homotopy Type of the Space of Diffeomorphisms. I
transactions of the american mathematical society Volume 196, 1974 THE HOMOTOPYTYPE OF THE SPACE OF DIFFEOMORPHISMS.I by dan burghelea(l) and richard lashof(2) ABSTRACT. A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result Diff(Dn, 3 ) ä Bn+ (Top /0 ) is obtained. Themain technique is fibrewise smoothing, n n 0. Introduction. Throughout this paper "differentiable" means C°° differenti- able. By a diffeomorphism, homeomorphism, pi homeomorphism, pd homeomor- phism /: M —»M, identity on K, we will mean a diffeomorphism, homeomorphism, etc., which is homotopic to the identity among continuous maps h: (¡A, dM) —* (M, dM), identity in K. Given M a differentiable compact manifold, one considers Diff(Mn, K) and Homeo(M", K) the topological groups of all diffeomorphisms (homeomorphisms) which are the identity on K endowed with the C00- (C )-topology. If K is a compact differentiable submanifold with or without boundary (possi- bly empty), Diff(Mn, K) is a C°°-differentiable Frechet manifold, hence (a) Diff(M", K) has the homotopy type of a countable CW-complex, (b) as a topological space Diff(M", K) is completely determined (up to home- omorphism) by its homotopy type (see [4] and [13j). In order to compare Diff(M", K) with Homeo(M", K) (as also with the com binatorial analogue of Homeo(Mn, K)), it is very convenient to consider the semi- simplicial version of Diff(* • •) and Homeo(- • •), namely, the semisimplicial groups Sd(Diff(Mn, K)) the singular differentiable complex of Diff(M", K) and S HomeoOW", K) the singular complex of Homeo(M'\ K). -
The Classifying Space of the G-Cobordism Category in Dimension Two
The classifying space of the G-cobordism category in dimension two Carlos Segovia Gonz´alez Instituto de Matem´aticasUNAM Oaxaca, M´exico 29 January 2019 The classifying space I Graeme Segal, Classifying spaces and spectral sequences (1968) I A. Grothendieck, Th´eorie de la descente, etc., S´eminaire Bourbaki,195 (1959-1960) I S. Eilenberg and S. MacLane, Relations between homology and homotopy groups of spaces I and II (1945-1950) I P.S. Aleksandrov-E.Chec,ˇ Nerve of a covering (1920s) I B. Riemann, Moduli space (1857) Definition A simplicial set X is a contravariant functor X : ∆ ! Set from the simplicial category to the category of sets. The geometric realization, denoted by jX j, is the topological space defined as ` jX j := n≥0(∆n × Xn)= ∼, where (s; X (f )a) ∼ (∆f (s); a) for all s 2 ∆n, a 2 Xm, and f :[n] ! [m] in ∆. Definition For a small category C we define the simplicial set N(C), called the nerve, denoted by N(C)n := Fun([n]; C), which consists of all the functors from the category [n] to C. The classifying space of C is the geometric realization of N(C). Denote this by BC := jN(C)j. Properties I The functor B : Cat ! Top sends a category to a topological space, a functor to a continuous map and a natural transformation to a homotopy. 0 0 I B conmutes with products B(C × C ) =∼ BC × BC . I A equivalence of categories gives a homotopy equivalence. I A category with initial or final object is contractil. -
Homological Algebra of Stable Homotopy Ring of Spheres
Pacific Journal of Mathematics HOMOLOGICAL ALGEBRA OF STABLE HOMOTOPY RING π∗ OF SPHERES T. Y. LIN Vol. 38, No. 1 March 1971 PACIFIC JOURNAL OF MATHEMATICS Vol. 38, No. 1, 1971 HOMOLOGICAL ALGEBRA OF STABLE HOMOTOPY RING ^ OF SPHERES T. Y. LIN The stable homotopy groups are studied as a graded ring π* via homoiogical algebra. The main object is to show that the projective (and weak) dimension of a finite type ^-module is oo unless the module is free. As a corollary, a partial answer to Whithead's corollary to Freyd's generating hy- pothesis is obtained. 1* Introduction and statement of main results* It is well- known that the stable homotopy groups of spheres form a commutative graded ring π* [20]. This paper is our first effort towards the investigation of the homoiogical properties of the stable homotopy ring π%. In this paper we have completed the computations of all the homoiogical numerical invariants of finitely generated type. The nonfinitely generated type will be taken up in forth-coming papers. The paper is organized as follows: The introduction is § 1. In § 2 we give a brief exposition about the theory of graded rings, which are needed in later sections. § 3 is primarily a preparation for § 4. Here the finitistic global dimensions of the "p-primary component" Λp of 7Γ* (precisely, Λp is a ring obtained by localizing π* at a maximal ideal) are computed, and a geometric realization of Λp is constructed. § 4 is the mainbody of this paper, here we prove Theorem 2 and derive from it Theorems 1, 3, and 4.