DOCSLIB.ORG

Blockseminar Constructions of Aspherical Manifolds and Complexes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Blockseminar Constructions of Aspherical Manifolds and Complexes Blockseminar Constructions of aspherical manifolds and complexes Organizers: Alexander Engel, Micha lMarcinkowski, Christopher Wulff Date: 24.-28. September 2018 Place: Frauenchiemsee Symmetric spaces The main reference for this part is [10]. Talk 1 (Lie groups and Lie algebras, and the exponential maps). Define Lie groups and Lie algebras, define the adjoint representation ad(−) and the center of a Lie algebra (p. 98{100). Discuss several examples of Lie groups and their Lie algebras (use references of your choice). Give the definition of the exponential map exp(−) from p. 104 (and discuss what you need from p. 102f) and of the exponential map Exp(−) from p. 33. Finally, prove Theorem 1.7 from p. 105 (for this you will have to discuss some things from x6 in Chapter I). Talk 2 (Semi-simple Lie algebras and the Cartan decomposition). Define the adjoint group Int(−), the automorphism group Aut(−) and the adjoint representation Ad(−) of a Lie group (p. 126f). Define compactly imbedded subalgebras and compact Lie algebras (p. 130). Define Killing forms and semi-simple Lie algebras (p, 131) and prove Propo- sition 6.6 (p. 132). State Proposition 6.10 without proof (p. 135). Give examples of semi-simple Lie algebras (use references of your choice). Discuss complexifications of Lie algebras and real forms (p. 180), state Theorem 6.3 (leave out the proof since we do not discuss roots in this seminar) and discuss Theorem 7.1 (p. 181f). Define Cartan decompositions (p. 183) and conclude from Theorems 6.3 and 7.1 that they always exist for semi-simple Lie algebras over R. Talk 3 (Symmetric spaces of non-compact type). You have to discuss many parts of x3 in Chapter IV. Start with the definition of a Riemannian globally symmetric space (p. 205), discuss Lemma 3.1 and state Lemma 3.2 (without proof). Discuss from Theorem 3.3 only the things which you need later. Define what a (Riemannian) symmetric pair is (p. 209) and what an orthogonal symmetric Lie algebra is (p. 213), and discuss Proposition 3.6. Discuss Theorem 4.1 (p. 215) and prove finally Theorem 1.1(iii) of Chapter IV. Aspherical presentation complexes Talk 4 (Small cancellation and one-relator groups). Prove that the presentation complex of a small cancellation group is aspherical 1 [14, Thm. 1(2)]. For this you will have to relate the definition of an aspherical presentation from [14] to asphericity of the presentation complex ([9, p. 32]) and show (at least sketch) [14, Lem. 13]. Prove that the presentation complex of a torsion-free one-relator group is aspherical [16, Thm. 7.1]. Reflection group trick Talk 5 (Coxeter groups). Define Coxeter groups by presentation ([3, Def.3.3.2], ignore pre-Coxeter systems). Give examples of Coxeter groups: dihedral groups [3, x3.1], your favourite example from [3, x6.5], discuss the 3-dimensional example given by the tiling of H3 by the right angled dodecahedron ([18, Ex.1.4.5] for existence of such a dodecahedron, if you have doubts about this construction a look at [3, Thm.6.4.3] might be helpful). Define the canonical representation [3, x6.12]. Note that the 3-dimensional example above may be used to construct closed 3-dimensional hyperbolic manifolds (by Selbergs' lemma every Coxeter group is virtually torsion free). State [3, Lem.4.7.2] and [3, Lem.4.7.3] (these combinatorial lemmas will be needed for the next talk). If time permits give ideas of the proofs. Talk 6 (Basic construction). Define U(G; X) assuming that G = (W; S) is a Coxeter group [3, x5.1]. Discuss the nerve of a Coxeter system [3, x7.1] and the complex U(W; K)[3, x7.2]. (focus on the last paragraph before Thm.7.2.4, you don't have to introduce abstract simplicial complexes and flags). Sketch the proof of [3, Thm.8.1.2] and [3, Thm.9.1.4]. Talk 7 (Aspherical manifolds not covered by Rn). Proof [3, Thm.10.5.1]. You have to explain in details the crucial Remark 10.5.2. To do that, you need to look at the proof of [3, Thm.9.2.2]. Focus rather on the fundamental group at infinity, not on semi-simplicity. Our examples are one-ended, thus while defining the fundamental group at infinity specify the discussion to the one-ended case. If time permits, give a concrete example of a homology sphere (e.g. [3, Ex.10.3.2]). Talk 8 (Aspherical manifolds not homotopy equivalent to smooth manifolds). Describe the reflection group trick: cover [3, x11.1] without Prop.11.1.3. As examples discuss shortly [3, Ex.11.2.1,Ex.11.2.2]. The main part of the talk is to prove [3, Thm.11.3.2] (see as well [4]). Give background on homotopy theory: explain what is an infinity loop space, classifying space and Spivak fibration (see [3, App.H2] and your favourite book on homotopy theory). CAT(0) geometry Talk 9 (CAT(κ) spaces and Gromov's version of the Cartan{Hadamard theorem). Give the definition of CAT(κ) spaces ([1, II.1]), discuss examples from [1, II.1.15] and prove [1, Thm. 4.1]. Talk 10 (CAT(0) cube complexes). Define cubical complexes [1, Thm. I.7.32] and prove Gromov's link condition for non-positive curvature [1, Thm. II.5.20]. 2 Examples: Salvetti complex for right angled Artin groups [2, x3.6], U(W; K) for right angled Coxeter group W [3, x12.2]. If time permits, discuss K(π; 1) conjecture for general Artin groups, [13, Def.4.5] and [13, Def.4.7]. Asphericalization `ala Kan{Thurston Talk 11 (Theorem of Kan{Thurston). Discuss and prove the Theorem of Kan{Thurston (you can ignore the statements about uniqueness of the spaces and about the Quillen +-construction). You can use either the original paper [11] or the article [12]. Hyperbolisation of polyhedra The reference for this part is [5]. Talk 12 (The Williams functor). Discuss thoroughly Section 1. Talk 13 (Main theorem and first application). Discuss Section 4 and prove the Main Theorem (4a.3). Explain the construction of hyperbolized simplices as needed in the main theorem (this is the main part of Section 4). As an application prove Theorem A from the introduction of the paper. Discuss also Theorem B. Talk 14 (Non-triangulable aspherical manifolds). Discuss Section 5a, i.e., Theorem (5a.1) and Corollary (5a.4). Explain as much as possible about the manifold E8 and the algebraic topology you need in the proof (use references of your choice), such that the audience can follow your exposition. Explain the plumbing construction for the construction of the manifold E8. More aspherical spaces Talk 15 (Universal realisation of cycles). Explain Steenrod's realisation problem, [8, x1] & [6, x1] (you can discuss Sullivan's resolution of singularities [6, x1] and the idea of the construction [6, x2]). The goal is to prove [8, Thm. 2.5]. If time permits discuss [7, Cor. 1.6]. Talk 16 (Aspherical manifolds and expanders). Sketch the construction of groups containing an isometric copy of an expander [15, Thm. 4] (some definitions concerning small cancellation theory are explained in Talk 4). Cover [15, x3.2.2] and state [17, Thm. 1.1] without proof. References [1] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, vol- ume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. [2] R. Charney. An introduction to right-angled Artin groups. Geom. Dedicata, 125:141{158, 2007. 3 [3] M. W. Davis. The geometry and topology of Coxeter groups, volume 32 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2008. [4] M. W. Davis and J.-C. Hausmann. Aspherical manifolds without smooth or PL structure. In Algebraic topology (Arcata, CA, 1986), volume 1370 of Lecture Notes in Math., pages 135{142. Springer, Berlin, 1989. [5] M. W. Davis and T. Januszkiewicz. Hyperbolization of polyhedra. J. Differential Geom., 34(2):347{388, 1991. [6] A. A. Gaifullin. Combinatorial realisation of cycles and small covers. In European Congress of Mathematics, pages 315{330. EMS, 2013. [7] A. A. Gaifullin. Universal realisators for homology classes. Geom. Topol., 17(3):1745{1772, 2013. [8] A. A. Gaifullin. Small covers over graph-associahedra and realization of cycles. Mat. Sb., 207(11):53{81, 2016. [9] S. M. Gersten, editor. Essays in group theory, volume 8 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1987. [10] S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces, volume 34 of Graduate Studies in Mathematics. American Mathematical Society, 2001. Reprint of the 1978 edition. [11] D. M. Kan and W. P. Thurston. Every connected space has the homology of a K(π; 1). Topology, 15(3):253{258, 1976. [12] C. R. F. Maunder. A short proof of a theorem of Kan and Thurston. Bull. London Math. Soc., 13(4):325{327, 1981. [13] J. McCammond. The mysterious geometry of Artin groups. http://web.math.ucsb.edu/~jon.mccammond/papers/mysterious-geometry.pdf. [14] Y. Ollivier. On a small cancellation theorem of Gromov. Bull. Belg. Math. Soc. Simon Stevin, 13(1):75{89, 2006. [15] D. Osajda. Small cancellation labellings of some infinite graphs and appli- cations. arxiv:1406.5015. [16] A. Putman. One-relator groups. https://www3.nd.edu/~andyp/notes/OneRelator.pdf. [17] M. Sapir. A Higman embedding preserving asphericity. J. Amer. Math. Soc., 27(1):1{42, 2014. [18] W. P. Thurston. Three-dimensional geometry and topology.
Load more

Recommended publications
  • NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last Updated: June 29, 2012
    Cube complexes Winter 2011 NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last updated: June 29, 2012 1 Some basics of topological and geometric group theory 1.1 Presentations and complexes Let Γ be a discrete group, defined by a presentation P = hai j rji, say, or as the fundamental group of a connected CW-complex X. 0 1 Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell Ei 2 for each ai (oriented accordingly), and one 2-cell Ej for each rj, with attaching 2 (1) map @Ej ! XP that reads off the word rj in the generators faig. Then, by the Seifert{van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X(2) and contracting a maximal tree in (1) X , we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a pre- sentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ. ∗ Question 1.2. Can we compute the homology H∗(Γ) or cohomology H (Γ)? Question 1.3. Can we solve the word problem in Γ? That is, is there an algorithm to determine whether a word in the generators represents the trivial element in Γ? 1.2 Non-positively curved manifolds In general, the answer to both of these questions is `no'. Novikov (1955) and Boone (1959) constructed a finitely presentable group with an unsolvable word problem, and Cameron Gordon proved that the rank of H2(Γ) is not computable.
    [Show full text]
  • Fundamental Groups of Links of Isolated Singularities
    FUNDAMENTAL GROUPS OF LINKS OF ISOLATED SINGULARITIES MICHAEL KAPOVICH AND JANOS´ KOLLAR´ Starting with Grothendieck's proof of the local version of the Lefschetz hyper- plane theorems [Gro68], it has been understood that there are strong parallels between the topology of smooth projective varieties and the topology of links of isolated singularities. This relationship was formulated as one of the guiding prin- ciples in the monograph [GM88, p.26]: \Philosophically, any statement about the projective variety or its embedding really comes from a statement about the sin- gularity at the point of the cone. Theorems about projective varieties should be consequences of more general theorems about singularities which are no longer re- quired to be conical". The aim of this note is to prove the following, which we consider to be a strong exception to this principle. Theorem 1. For every finitely presented group G there is an isolated, 3-dimensi- ∼ onal, complex singularity 0 2 XG with link LG such that π1 LG = G. By contrast, the fundamental groups of smooth projective varieties are rather special; see [ABC+96] for a survey. Even the fundamental groups of smooth quasi projective varieties are quite restricted [Mor78, KM98a, CS08, DPS09]. This shows that germs of singularities can also be quite different from quasi projective varieties. We think of a complex singularity (0 2 X) as a contractible Stein space sitting N 2N−1 in some C . Then its link is link(X) := X \ S , an intersection of X with a small (2N − 1)-sphere centered at 0 2 X. Thus link(X) is a deformation retract of X n f0g.
    [Show full text]
  • Lectures on Representations of Surface Groups François Labourie
    Lectures on Representations of Surface Groups François Labourie To cite this version: François Labourie. Lectures on Representations of Surface Groups. EMS publishing house, pp.145, 2013, Zurich Lectures in Advanced Mathematics, 978-3-03719-127-9. hal-00627916 HAL Id: hal-00627916 https://hal.archives-ouvertes.fr/hal-00627916 Submitted on 29 Sep 2011 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. Copyright Lectures on Representations of Surface Groups Notes of a course given in ETH-Z¨urich Fall 2006, and Orsay Spring 2007-2008. Fran¸coisLABOURIE 1 (Based on notes taken by Tobias HARTNICK) April 12, 2011 1Univ. Paris-Sud, Laboratoire de Math´ematiques,Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405, The research leading to these results has received funding from the European Research Council under the European Community's seventh Frame- work Programme (FP7/2007-2013)/ERC grant agreement n◦ FP7-246918, as well as well as from the ANR program ETTT (ANR-09-BLAN-0116-01) Chapter 1 Introduction The purpose of this lecture series is to describe the representation variety { or character variety{ of the fundamental group π1(S) of a closed connected surface S of genus greater than 2, with values in a Lie group G.
    [Show full text]
  • H. Wilton, Non-Positively Curved Cube Complexes
    Cube complexes Winter 2011 NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last updated: March 8, 2011 1 Some basics of topological and geometric group theory 1.1 Presentations and complexes Let Γ be a discrete group, defined by a presentation P = hai j rji, say, or as the fundamental group of a connected CW-complex X. 0 1 Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell Ei 2 for each ai (oriented accordingly), and one 2-cell Ej for each rj, with attaching 2 (1) map @Ej ! XP that reads off the word rj in the generators faig. Then, by the Seifert{van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X(2) and contracting a maximal tree in (1) X , we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a pre- sentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ. ∗ Question 1.2. Can we compute the homology H∗(Γ) or cohomology H (Γ)? Question 1.3. Can we solve the word problem in Γ? That is, is there an algorithm to determine whether a word in the generators represents the trivial element in Γ? 1.2 Non-positively curved manifolds In general, the answer to both of these questions is `no'. Novikov (1955) and Boone (1959) constructed a finitely presentable group with an unsolvable word problem, and Cameron Gordon proved that the rank of H2(Γ) is not computable.
    [Show full text]
  • Iteration of Functions and Contractibility of Acyclic 2-Complexes
    Iteration of functions and contractibility of acyclic 2-complexes Ian J. Leary∗ March 23, 2020 Abstract We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds. 1 Introduction The existence of an algorithm to determine which finite 2-dimensional simplicial com- plexes are contractible is a well-known open problem. There are good algorithms to compute the homology of a finite complex, and so the problem quickly reduces to the case of acyclic complexes, i.e., those having the same homology as a point. This problem can be stated as a problem about finite presentations of groups: is there an algorithm to decide which finite balanced presentations of perfect groups are trivial? The problem was stated in this form and attributed to Magnus in the first edition of the Kourovka notebook [5, 1.12], and appears as problem (FP1) in [1]. A presentation is balanced if it has the same numbers of generators and relators. The conditions that the group be per- fect and that the presentation be balanced are equivalent to the corresponding 2-complex being acyclic. If a finite presentation presents the trivial group, then a systematic search will even- tually find a proof of this fact. Thus there is a partial algorithm that will verify that a finite simplicial complex (of any dimension) is contractible, but that will fail to halt in general. arXiv:1910.00931v3 [math.GR] 20 Mar 2020 In general there is no algorithm to decide whether a finite presentation describes the trivial group.
    [Show full text]
  • Iteration of Functions and Contractibility of Acyclic 2-Complexes
    Iteration of functions and contractibility of acyclic 2-complexes Ian J. Leary∗ March 20, 2020 Abstract We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds. 1 Introduction The existence of an algorithm to determine which finite 2-dimensional simplicial com- plexes are contractible is a well-known open problem. There are good algorithms to compute the homology of a finite complex, and so the problem quickly reduces to the case of acyclic complexes, i.e., those having the same homology as a point. This problem can be stated as a problem about finite presentations of groups: is there an algorithm to decide which finite balanced presentations of perfect groups are trivial? The problem was stated in this form and attributed to Magnus in the first edition of the Kourovka notebook [5, 1.12], and appears as problem (FP1) in [1]. A presentation is balanced if it has the same numbers of generators and relators. The conditions that the group be per- fect and that the presentation be balanced are equivalent to the corresponding 2-complex being acyclic. If a finite presentation presents the trivial group, then a systematic search will even- tually find a proof of this fact. Thus there is a partial algorithm that will verify that a finite simplicial complex (of any dimension) is contractible, but that will fail to halt in general. In general there is no algorithm to decide whether a finite presentation describes the trivial group.
    [Show full text]
  • Introduction to Algebraic Topology
    Introduction to Algebraic Topology Johnny Nicholson https://www.ucl.ac.uk/~ucahjni/ Abstract These notes are written to accompany the lecture course `Introduction to Algebraic Topology' that was taught to advanced high school students during the Ross Mathematics Program in Columbus, Ohio from July 15th-19th, 2019. The course was taught over five lectures of 1-1.5 hours and the students were assumed to have some experience working with groups and rings, as well as some familiarity with the definition of a metric space and a continuous function. These notes were adapted from the notes taken down by Alex Feiner and Ojaswi Acharya. Special thanks go to Alex for creating most of the diagrams. We would also like to thank Evan O'Dorney for comments and corrections. Further comments and corrections may be sent to [email protected]. Contents 1 Spaces and Equivalences1 1.1 Notions of Space...........................................1 1.1.1 Subsets of Rn .........................................1 1.1.2 Finite Cell Complexes....................................2 1.2 Notions of Equivalence........................................4 1.2.1 Homeomorphism.......................................4 1.2.2 Homotopy Equivalence....................................5 2 Invariants 8 2.1 The Euler Characteristic.......................................8 2.2 The Fundamental Group.......................................9 2.2.1 Homotopy Invariance.................................... 11 2.2.2 The Fundamental Group of S1 ............................... 14 2.3 Other Invariants........................................... 16 3 Applications 19 3.1 The Fundamental Theorem of Algebra............................... 19 3.2 Brouwer Fixed Point Theorem................................... 20 4 Classification Problems 22 4.1 Classification of 1-Complexes.................................... 22 4.2 Classification of 2-Complexes.................................... 23 4.2.1 Group Presentations....................................
    [Show full text]
  • Lectures on Geometric Group Theory
    Lectures on Geometric Group Theory Cornelia Drutu and Michael Kapovich Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov’s theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz’s quasi-isometric rigidity theorem for nonuniform lattices in the real-hyperbolic spaces. We conclude with a survey of the quasi-isometric rigidity theory. The main idea of the geometric group theory is to treat finitely-generated groups as geometric objects: With each finitely-generated group G one associates a metric space, the Cayley graph of G. One of the main issues of the geometric group theory is to recover as much as possible algebraic information about G from the geometry of the Cayley graph. (A somewhat broader viewpoint is to say that one studies a finitely generated group G by analyzing geometric properties of spaces X on which G acts geometrically, i.e., properly discontinuously, cocompactly and isometrically. The Cayley graph is just one of such spaces.) A primary obstacle for this is the fact that the Cayley graph depends not only on G but on a particular choice of a generating set of G. Cayley graphs associated with different generating sets are not isometric but quasi-isometric. The fundamental question which we will try to address in this book is: If G; G0 are quasi-isometric groups, to which extent G and G0 share the same algebraic properties? The best one can hope here is to recover the group G up to virtual isomorphism from its geometry.
    [Show full text]
  • Part IV — Topics in Geometric Group Theory Definitions
    Part IV | Topics in Geometric Group Theory Definitions Based on lectures by H. Wilton Notes taken by Dexter Chua Michaelmas 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. The subject of geometric group theory is founded on the observation that the algebraic and algorithmic properties of a discrete group are closely related to the geometric features of the spaces on which the group acts. This graduate course will provide an introduction to the basic ideas of the subject. Suppose Γ is a discrete group of isometries of a metric space X. We focus on the theorems we can prove about Γ by imposing geometric conditions on X. These conditions are motivated by curvature conditions in differential geometry, but apply to general metric spaces and are much easier to state. First we study the case when X is Gromov-hyperbolic, which corresponds to negative curvature. Then we study the case when X is CAT(0), which corresponds to non-positive curvature. In order for this theory to be useful, we need a rich supply of negatively and non-positively curved spaces. We develop the theory of non-positively curved cube complexes, which provide many examples of CAT(0) spaces and have been the source of some dramatic developments in low-dimensional topology over the last twenty years. Part 1. We will introduce the basic notions of geometric group theory: Cayley graphs, quasiisometries, the Schwarz{Milnor Lemma, and the connection with algebraic topology via presentation complexes.
    [Show full text]
  • Course Portfolio for Math 872: Topology II Alex Zupan
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln UNL Faculty Course Portfolios Peer Review of Teaching Project 2019 Course Portfolio for Math 872: Topology II Alex Zupan Follow this and additional works at: https://digitalcommons.unl.edu/prtunl Part of the Higher Education Commons, and the Higher Education and Teaching Commons This Portfolio is brought to you for free and open access by the Peer Review of Teaching Project at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in UNL Faculty Course Portfolios by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Course Portfolio for Math 872: Topology II Promoting graduate student engagement Alex Zupan Department of Mathematics University of Nebraska{Lincoln [email protected] May 2019 Abstract In this course portfolio, I explore tactics for engaging a group of beginning mathemat- ics graduate students taking the second semester of the first-year sequence in topology. Course objectives include computing algebraic invariants of topological spaces, under- standing the Galois correspondence between covering spaces and fundamental groups, and using a range of invariants to distinguish spaces. My main instructional tools include a series of three \Activities" tailored to each student, a project involving com- puting invariants with the program SnapPy and using homology of covering spaces to distinguish 3-dimensional knots, and individual student board work during class periods. The activities served as benchmarks to gauge student understanding of key concepts; the first functioned as a pretest, the second was completed mid-semester, and the third served as a posttest. I have used the activities to analyze the progres- sion of student understanding over the course of the semester, and in addition, I have examined student performance on midterm exams in order to determine the extent to which individual board work impacted learning.
    [Show full text]
  • Hyperbolic Groups Lecture Notes∗
    Hyperbolic groups Lecture Notes∗ M. Hull 1 Geometric group theory basics Word metric and Cayley graphs Let G be a group generated by S ⊆ G; for convenience, we will always assume that our generating sets are symmetric, that is S = S−1.A word in S is a finite concatenation of elements of S. For such a word W , let kW k denote its length. If two words W and U are letter for letter equivalent, we write W ≡ U, and if W and U represent the same element of the group G, we write W =G U. For an element g 2 G, let jgjS denote the length of the shortest −1 word in S which represents g in the group G. Given g; h 2 G, let dS(g; h) = jg hjS. dS is called the word metric on G with respect to S. We let Γ(G; S) denote the Cayley graph of G with respect to S. This is the graph whose vertex set is G and there is an oriented edge e labeled by s 2 S between any two vertices of the form g and gs. We typically identify the edges labeled by s and s−1 with the same endpoints and consider these as the same edge with opposite orientations. Lab(e) denotes the label of the edge e; similarly, for a (combinatorial) path p, Lab(p) will denote the concatenation of the labels of the edges of p. Also for such a path p, we let p− and p+ denote the intial and the terminal vertex of p respectively, and `(p) will denote the number of edges of p.
    [Show full text]
  • Math 525 Algebraic Topology I. Spring 2021. Igor Mineyev Homework, Topics, Projects, and Fun
    Math 525 Algebraic topology I. Spring 2021. Igor Mineyev Homework, topics, projects, and fun. Below, \∗" means \turn in", \no ∗" means \do not turn in, but know how to solve". If a text is in yellow color, the homework is still at a preliminary stage and might be modified later, but feel free to start working on it. The problems marked \for extra fun" are some interesting related problems; they will not affect your grade for the course, but should be good sources of inspiration. I will also include a list of topics. Topics: Metric space, topology, open sets, closed sets, topological space, examples of topo- logical spaces, continuous function (= map), homeomorphism; constructing new topological spaces: subspace topology, the topology of disjoint union, product topology, quotient topology; disk Dn, boundary of a disk, manifold, sphere Sn, torus T n, projective plane (3 definitions), projective space, examples of surfaces, attaching map, cell complex (= CW-complex), weak topology, cellular structures on Sn and RPn, paths and loops in a topological space, homotopy of maps, path homotopy, path homotopy is an equivalence relation, concatenation of paths, n loops, fundamental group, π1(R ), path-connected topological space, simply connected, change of basepoint, ... Homework 1. Due before 2pm on Friday, February 5. (1*) Give three different definitions of the projective plane RP2 (as in class). Prove that they give the same topological space (i.e. they are homeomorphic). Generalize to RPn. (2*) Prove that RP2 is a manifold. (Do not forget Hausdorff.) (3*) Let X be the result of collapsing @D2 in the disk D2 to a point, with the quotient topology.
    [Show full text]