Homotopies, Loops, and the Fundamental Group 1 1 2
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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. -
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). -
Lens Spaces We Introduce Some of the Simplest 3-Manifolds, the Lens Spaces
CHAPTER 10 Seifert manifolds In the previous chapter we have proved various general theorems on three- manifolds, and it is now time to construct examples. A rich and important source is a family of manifolds built by Seifert in the 1930s, which generalises circle bundles over surfaces by admitting some “singular”fibres. The three- manifolds that admit such kind offibration are now called Seifert manifolds. In this chapter we introduce and completely classify (up to diffeomor- phisms) the Seifert manifolds. In Chapter 12 we will then show how to ge- ometrise them, by assigning a nice Riemannian metric to each. We will show, for instance, that all the elliptic andflat three-manifolds are in fact particular kinds of Seifert manifolds. 10.1. Lens spaces We introduce some of the simplest 3-manifolds, the lens spaces. These manifolds (and many more) are easily described using an important three- dimensional construction, called Dehnfilling. 10.1.1. Dehnfilling. If a 3-manifoldM has a spherical boundary com- ponent, we can cap it off with a ball. IfM has a toric boundary component, there is no canonical way to cap it off: the simplest object that we can attach to it is a solid torusD S 1, but the resulting manifold depends on the gluing × map. This operation is called a Dehnfilling and we now study it in detail. LetM be a 3-manifold andT ∂M be a boundary torus component. ⊂ Definition 10.1.1. A Dehnfilling ofM alongT is the operation of gluing a solid torusD S 1 toM via a diffeomorphismϕ:∂D S 1 T. -
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. -
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. -
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. -
A Computation of Knot Floer Homology of Special (1,1)-Knots
A COMPUTATION OF KNOT FLOER HOMOLOGY OF SPECIAL (1,1)-KNOTS Jiangnan Yu Department of Mathematics Central European University Advisor: Andras´ Stipsicz Proposal prepared by Jiangnan Yu in part fulfillment of the degree requirements for the Master of Science in Mathematics. CEU eTD Collection 1 Acknowledgements I would like to thank Professor Andras´ Stipsicz for his guidance on writing this thesis, and also for his teaching and helping during the master program. From him I have learned a lot knowledge in topol- ogy. I would also like to thank Central European University and the De- partment of Mathematics for accepting me to study in Budapest. Finally I want to thank my teachers and friends, from whom I have learned so much in math. CEU eTD Collection 2 Abstract We will introduce Heegaard decompositions and Heegaard diagrams for three-manifolds and for three-manifolds containing a knot. We define (1,1)-knots and explain the method to obtain the Heegaard diagram for some special (1,1)-knots, and prove that torus knots and 2- bridge knots are (1,1)-knots. We also define the knot Floer chain complex by using the theory of holomorphic disks and their moduli space, and give more explanation on the chain complex of genus-1 Heegaard diagram. Finally, we compute the knot Floer homology groups of the trefoil knot and the (-3,4)-torus knot. 1 Introduction Knot Floer homology is a knot invariant defined by P. Ozsvath´ and Z. Szabo´ in [6], using methods of Heegaard diagrams and moduli theory of holomorphic discs, combined with homology theory. -
Determining Hyperbolicity of Compact Orientable 3-Manifolds with Torus
Determining hyperbolicity of compact orientable 3-manifolds with torus boundary Robert C. Haraway, III∗ February 1, 2019 Abstract Thurston’s hyperbolization theorem for Haken manifolds and normal sur- face theory yield an algorithm to determine whether or not a compact ori- entable 3-manifold with nonempty boundary consisting of tori admits a com- plete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm. 1 Introduction The work of Jørgensen, Thurston, and Gromov in the late ‘70s showed ([18]) that the set of volumes of orientable hyperbolic 3-manifolds has order type ωω. Cao and Meyerhoff in [4] showed that the first limit point is the volume of the figure eight knot complement. Agol in [1] showed that the first limit point of limit points is the volume of the Whitehead link complement. Most significantly for the present paper, Gabai, Meyerhoff, and Milley in [9] identified the smallest, closed, orientable hyperbolic 3-manifold (the Weeks-Matveev-Fomenko manifold). arXiv:1410.7115v5 [math.GT] 31 Jan 2019 The proof of the last result required distinguishing hyperbolic 3-manifolds from non-hyperbolic 3-manifolds in a large list of 3-manifolds; this was carried out in [17]. The method of proof was to see whether the canonize procedure of SnapPy ([5]) succeeded or not; identify the successes as census manifolds; and then examine the fundamental groups of the 66 remaining manifolds by hand. This method made the ∗Research partially supported by NSF grant DMS-1006553. -
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. -
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 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). -
Cohomology Determinants of Compact 3-Manifolds
COHOMOLOGY DETERMINANTS OF COMPACT 3–MANIFOLDS CHRISTOPHER TRUMAN Abstract. We give definitions of cohomology determinants for compact, connected, orientable 3–manifolds. We also give formu- lae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology de- terminants are related to Turaev torsion, though the author hopes that they have other uses as well. 1. Introduction Cohomology determinants are an invariant of compact, connected, orientable 3-manifolds. The author first encountered these invariants in [Tur02], when Turaev gave the definition for closed 3–manifolds, and used cohomology determinants to obtain a leading order term of Tu- raev torsion. In [Tru], the author gives a definition for 3–manifolds with boundary, and derives a similar relationship to Turaev torsion. Here, we repeat the definitions, and give formulae relating the cohomology determinants before and after gluing a solid torus along a boundary component. One can use these formulae, and gluing formulae for Tu- raev torsion from [Tur02] Chapter VII, to re-derive the results of [Tru] from the results of [Tur02] Chapter III, or vice-versa. 2. Integral Cohomology Determinants arXiv:math/0611248v1 [math.GT] 8 Nov 2006 2.1. Closed 3–manifolds. We will simply state the relevant result from [Tur02] Section III.1; the proof is similar to the one below. Let R be a commutative ring with unit, and let N be a free R–module of rank n ≥ 3. Let S = S(N ∗) be the graded symmetric algebra on ∗ ℓ N = HomR(N, R), with grading S = S . Let f : N ×N ×N −→ R ℓL≥0 be an alternate trilinear form, and let g : N × N → N ∗ be induced by f.