Survey on Classifying Spaces for Families of Subgroups
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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. -
1. CW Complex 1.1. Cellular Complex. Given a Space X, We Call N-Cell a Subspace That Is Home- Omorphic to the Interior of an N-Disk
1. CW Complex 1.1. Cellular Complex. Given a space X, we call n-cell a subspace that is home- omorphic to the interior of an n-disk. Definition 1.1. Given a Hausdorff space X, we call cellular structure on X a n collection of disjoint cells of X, say ffeαgα2An gn2N (where the An are indexing sets), together with a collection of continuous maps called characteristic maps, say n n n k ffΦα : Dα ! Xgα2An gn2N, such that, if X := feαj0 ≤ k ≤ ng (for all n), then : S S n (1) X = ( eα), n2N α2An n n n (2)Φ α Int(Dn) is a homeomorphism of Int(D ) onto eα, n n F k (3) and Φα(@D ) ⊂ eα. k n−1 eα2X We shall call X together with a cellular structure a cellular complex. n n n Remark 1.2. We call X ⊂ feαg the n-skeleton of X. Also, we shall identify X S k n n with eα and vice versa for simplicity of notations. So (3) becomes Φα(@Dα) ⊂ k n eα⊂X S Xn−1. k k n Since X is a Hausdorff space (in the definition above), we have that eα = Φα(D ) k n k (where the closure is taken in X). Indeed, we easily have that Φα(D ) ⊂ eα by n n continuity of Φα. Moreover, since D is compact, we have that its image under k k n k Φα also is, and since X is Hausdorff, Φα(D ) is a closed subset containing eα. Hence we have the other inclusion (Remember that the closure of a subset is the k intersection of all closed subset containing it). -
Homotopy Theory of Diagrams and Cw-Complexes Over a Category
Can. J. Math.Vol. 43 (4), 1991 pp. 814-824 HOMOTOPY THEORY OF DIAGRAMS AND CW-COMPLEXES OVER A CATEGORY ROBERT J. PIACENZA Introduction. The purpose of this paper is to introduce the notion of a CW com plex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category. A brief description of the paper goes as follows: in Section 1 we introduce the homo topy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type ho motopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method. 1. Homotopy theory of diagrams. Throughout this paper we let Top be the carte sian closed category of compactly generated spaces in the sense of Vogt [10]. -
1 CW Complex, Cellular Homology/Cohomology
Our goal is to develop a method to compute cohomology algebra and rational homotopy group of fiber bundles. 1 CW complex, cellular homology/cohomology Definition 1. (Attaching space with maps) Given topological spaces X; Y , closed subset A ⊂ X, and continuous map f : A ! y. We define X [f Y , X t Y / ∼ n n n−1 n where x ∼ y if x 2 A and f(x) = y. In the case X = D , A = @D = S , D [f X is said to be obtained by attaching to X the cell (Dn; f). n−1 n n Proposition 1. If f; g : S ! X are homotopic, then D [f X and D [g X are homotopic. Proof. Let F : Sn−1 × I ! X be the homotopy between f; g. Then in fact n n n D [f X ∼ (D × I) [F X ∼ D [g X Definition 2. (Cell space, cell complex, cellular map) 1. A cell space is a topological space obtained from a finite set of points by iterating the procedure of attaching cells of arbitrary dimension, with the condition that only finitely many cells of each dimension are attached. 2. If each cell is attached to cells of lower dimension, then the cell space X is called a cell complex. Define the n−skeleton of X to be the subcomplex consisting of cells of dimension less than n, denoted by Xn. 3. A continuous map f between cell complexes X; Y is called cellular if it sends Xk to Yk for all k. Proposition 2. 1. Every cell space is homotopic to a cell complex. -
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]. -
Generalized Cohomology Theories
Lecture 4: Generalized cohomology theories 1/12/14 We've now defined spectra and the stable homotopy category. They arise naturally when considering cohomology. Proposition 1.1. For X a finite CW-complex, there is a natural isomorphism 1 ∼ r [Σ X; HZ]−r = H (X; Z). The assumption that X is a finite CW-complex is not necessary, but here is a proof in this case. We use the following Lemma. Lemma 1.2. ([A, III Prop 2.8]) Let F be any spectrum. For X a finite CW- 1 n+r complex there is a natural identification [Σ X; F ]r = colimn!1[Σ X; Fn] n+r On the right hand side the colimit is taken over maps [Σ X; Fn] ! n+r+1 n+r [Σ X; Fn+1] which are the composition of the suspension [Σ X; Fn] ! n+r+1 n+r+1 n+r+1 [Σ X; ΣFn] with the map [Σ X; ΣFn] ! [Σ X; Fn+1] induced by the structure map of F ΣFn ! Fn+1. n+r Proof. For a map fn+r :Σ X ! Fn, there is a pmap of degree r of spectra Σ1X ! F defined on the cofinal subspectrum whose mth space is ΣmX for m−n−r m ≥ n+r and ∗ for m < n+r. This pmap is given by Σ fn+r for m ≥ n+r 0 n+r and is the unique map from ∗ for m < n+r. Moreover, if fn+r; fn+r :Σ X ! 1 Fn are homotopic, we may likewise construct a pmap Cyl(Σ X) ! F of degree n+r 1 r. -
Homology of Cell Complexes
Geometry and Topology of Manifolds 2015{2016 Homology of Cell Complexes Cell Complexes Let us denote by En the closed n-ball, whose boundary is the sphere Sn−1.A cell complex (or CW-complex) is a topological space X equipped with a filtration by subspaces X(0) ⊆ X(1) ⊆ X(2) ⊆ · · · ⊆ X(n) ⊆ · · · (1) (0) 1 (n) where X is discrete (i.e., has the discrete topology), such that X = [n=0X as a topological space, and, for all n, the space X(n) is obtained from X(n−1) by attaching n−1 (n−1) a set of n-cells by means of a family of maps f'j : S ! X gj2Jn . Here Jn is any set of indices and X(n) is therefore a quotient of the disjoint union X(n−1) ` En j2Jn n−1 (n−1) where we identify, for each point x 2 S , the image 'j(x) 2 X with the point x in the j-th copy of En. It is customary to denote (n) (n−1) n X = X [f'j g fej g n (n) and view each ej as an \open n-cell". The space X is called the n-skeleton of X, and the filtration (1) together with the attaching maps f'jgj2Jn for all n is called a cell decomposition or CW-decomposition of X. Thus a topological space may admit many distinct cell decompositions (or none). Cellular Chain Complexes (n) Let X be a cell complex with n-skeleton X and attaching maps f'jgj2Jn for all n. -
On the Non-Existence of Elements of Kervaire Invariant
ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Dedicated to Mark Mahowald 0 Abstract. We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology. Contents 1. Introduction 1 2. Equivariant stable homotopy theory 8 3. Mackey functors, homology and homotopy 35 4. The slice filtration 44 5. The complex cobordism spectrum 63 6. The Slice Theorem and the Reduction Theorem 76 7. The Reduction Theorem 80 8. The Gap Theorem 88 9. The Periodicity Theorem 89 10. The Homotopy Fixed Point Theorem 101 11. The Detection Theorem 103 Appendix A. The category of equivariant orthogonal spectra 118 Appendix B. Homotopy theory of equivariant orthogonal spectra 143 References 217 arXiv:0908.3724v4 [math.AT] 25 May 2015 1. Introduction The existence of smooth framed manifolds of Kervaire invariant one is one of the oldest unresolved issues in differential and algebraic topology. The question originated in the work of Pontryagin in the 1930’s. It took a definitive form in the paper [41] of Kervaire in which he constructed a combinatorial 10-manifold with no smooth structure, and in the work of Kervaire-Milnor [42] on h-cobordism classes M. A. Hill was partially supported by NSF grants DMS-0905160 , DMS-1307896 and the Sloan foundation. -
Algebraic K-Theory and Manifold Topology
Course Syllabus for Math 281: Algebraic K-theory and Manifold Topology Course Description Let M and N be smooth closed manifolds of dimension n. An h-cobordism from M to N is a compact smooth manifold B of dimension (n + 1) with boundary @B ' M q N having the property that the inclusion maps M,! B,! N are homotopy equivalences. If n ≥ 5 and the manifold M is simply connected, then the celebrated h-cobordism theorem of Smale asserts that B is diffeomorphic to a product M × [0; 1] (and, in particular, M is diffeomorphic to N). If M is not simply connected, then it is generally not true that any h-cobordism B from M to N is diffeomorphic to a product M × [0; 1]. In fact, one can introduce an algebraic invariant τ(B) (called the Whitehead torsion of B) belonging to an abelian group W h(M) (called the Whitehead group of M). The invariant τ(B) vanishes whenever B is diffeomorphic to a product M × [0; 1], and the converse holds provided that the dimension of M is greater than 5 (this statement is known as the s-cobordism theorem). The Whitehead group W h(M) depends only on the fundamental group of M and vanishes when M is simply connected, so that the s-cobordism theorem implies the h-cobordism theorem. For many purposes, it is useful to know whether there is an analogue of the s-cobordism theorem for families of manifolds. Fix a compact smooth n-dimensional manifold M, a finite cell complex X, and suppose we given a fiber bundle B ! X where each of the fibers Bx is an h-cobordism from M to some other n-manifold Nx (where the smooth structures vary continuously with x). -
A CARDINALITY FILTRATION Contents 1. Preliminaries 1 1.1
A CARDINALITY FILTRATION ARAMINTA AMABEL Contents 1. Preliminaries 1 1.1. Stratified Spaces 1 1.2. Exiting Disks 2 2. Recollections 2 2.1. Linear Poincar´eDuality 3 3. The Ran Space 5 4. Disk Categories 6 4.1. Cardinality Filtration 7 5. Layer Computations 9 5.1. Reduced Extensions 13 6. Convergence 15 References 16 1. Preliminaries 1.1. Stratified Spaces. Two talks ago, we discussed zero-pointed manifolds M∗. For parts of what we will do today, we will need to consider a wider class of spaces: (conically smooth) stratified spaces. Definition 1.1. Let P be a partially ordered set. Topologize P by declaring sets that are closed upwards to be open. A P -stratified space is a topological space X together with a continuous map −1 f : X ! P . We refer to the fiber f fpg =: Xp as the p-stratum. We view zero-pointed manifolds as stratified spaces over f0 < 1g with ∗ in the 0 stratum and M = M∗ n ∗ in the 1 stratum. Definition 1.2. A stratified space X ! P is called conically stratified if for every p 2 A and x 2 Xp there exists an P>p-stratified space U and a topological space Z together with an open embedding C(Z) × U ! X ` of stratified spaces whose image contains x. Here C(Z) = ∗ (Z × R≥0). Z×{0g We will not discuss the definition of conically smooth stratified space. For now, just think of conically smooth stratified spaces as conically stratified spaces that are nice and smooth like manifolds. Details can be found in [4]. -
The Unitary Group in Its Strong Topology
Advances in Pure Mathematics, 2018, 8, 508-515 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 The Unitary Group in Its Strong Topology Martin Schottenloher Mathematisches Institut, LMU München, Theresienstr 39, München, Germany How to cite this paper: Schottenloher, M. Abstract (2018) The Unitary Group in Its Strong Topology. Advances in Pure Mathematics, The goal of this paper is to confirm that the unitary group U() on an 8, 508-515. infinite dimensional complex Hilbert space is a topological group in its https://doi.org/10.4236/apm.2018.85029 strong topology, and to emphasize the importance of this property for Received: April 4, 2018 applications in topology. In addition, it is shown that U() in its strong Accepted: May 28, 2018 topology is metrizable and contractible if is separable. As an application Published: May 31, 2018 Hilbert bundles are classified by homotopy. Copyright © 2018 by author and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Unitary Operator, Strong Operator Topology, Topological Group, Infinite Commons Attribution International Dimensional Lie Group, Contractibility, Hilbert Bundle, Classifying Space License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction The unitary group U() plays an essential role in many areas of mathematics and physics, e.g. in representation theory, number theory, topology and in quantum mechanics. In some of the corresponding research articles complicated proofs and constructions have been introduced in order to circumvent the assumed fact that the unitary group is not a topological group when equipped with the strong topology (see Remark 1 below for details).