Lecture 3 TOPOLOGICAL CONSTRUCTIONS in This Lecture
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The Topology of Subsets of Rn
Lecture 1 The topology of subsets of Rn The basic material of this lecture should be familiar to you from Advanced Calculus courses, but we shall revise it in detail to ensure that you are comfortable with its main notions (the notions of open set and continuous map) and know how to work with them. 1.1. Continuous maps “Topology is the mathematics of continuity” Let R be the set of real numbers. A function f : R −→ R is called continuous at the point x0 ∈ R if for any ε> 0 there exists a δ> 0 such that the inequality |f(x0) − f(x)| < ε holds for all x ∈ R whenever |x0 − x| <δ. The function f is called contin- uous if it is continuous at all points x ∈ R. This is basic one-variable calculus. n Let R be n-dimensional space. By Or(p) denote the open ball of radius r> 0 and center p ∈ Rn, i.e., the set n Or(p) := {q ∈ R : d(p, q) < r}, where d is the distance in Rn. A function f : Rn −→ R is called contin- n uous at the point p0 ∈ R if for any ε> 0 there exists a δ> 0 such that f(p) ∈ Oε(f(p0)) for all p ∈ Oδ(p0). The function f is called continuous if it is continuous at all points p ∈ Rn. This is (more advanced) calculus in several variables. A set G ⊂ Rn is called open in Rn if for any point g ∈ G there exists a n δ> 0 such that Oδ(g) ⊂ G. -
The Suspension X ∧S and the Fake Suspension ΣX of a Spectrum X
Contents 41 Suspensions and shift 1 42 The telescope construction 10 43 Fibrations and cofibrations 15 44 Cofibrant generation 22 41 Suspensions and shift The suspension X ^S1 and the fake suspension SX of a spectrum X were defined in Section 35 — the constructions differ by a non-trivial twist of bond- ing maps. The loop spectrum for X is the function complex object 1 hom∗(S ;X): There is a natural bijection 1 ∼ 1 hom(X ^ S ;Y) = hom(X;hom∗(S ;Y)) so that the suspension and loop functors are ad- joint. The fake loop spectrum WY for a spectrum Y con- sists of the pointed spaces WY n, n ≥ 0, with adjoint bonding maps n 2 n+1 Ws∗ : WY ! W Y : 1 There is a natural bijection hom(SX;Y) =∼ hom(X;WY); so the fake suspension functor is left adjoint to fake loops. n n+1 The adjoint bonding maps s∗ : Y ! WY define a natural map g : Y ! WY[1]: for spectra Y. The map w : Y ! W¥Y of the last section is the filtered colimit of the maps g Wg[1] W2g[2] Y −! WY[1] −−−! W2Y[2] −−−! ::: Recall the statement of the Freudenthal suspen- sion theorem (Theorem 34.2): Theorem 41.1. Suppose that a pointed space X is n-connected, where n ≥ 0. Then the homotopy fibre F of the canonical map h : X ! W(X ^ S1) is 2n-connected. 2 In particular, the suspension homomorphism 1 ∼ 1 piX ! pi(W(X ^ S )) = pi+1(X ^ S ) is an isomorphism for i ≤ 2n and is an epimor- phism for i = 2n+1, provided that X is connected. -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Local Topological Properties of Asymptotic Cones of Groups
Local topological properties of asymptotic cones of groups Greg Conner and Curt Kent October 8, 2018 Abstract We define a local analogue to Gromov’s loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature. Contents 1 Introduction 1 1.1 Definitions...................................... 3 2 Coarse Loop Division Property 5 2.1 Absolutely non-divisible sequences . .......... 10 2.2 Simplyconnectedcones . .. .. .. .. .. .. .. 11 3 Examples 15 3.1 An example of a group with locally simply connected cones which is not simply connected ........................................ 17 1 Introduction Gromov [14, Section 5.F] was first to notice a connection between the homotopic properties of asymptotic cones of a finitely generated group and algorithmic properties of the group: if arXiv:1210.4411v1 [math.GR] 16 Oct 2012 all asymptotic cones of a finitely generated group are simply connected, then the group is finitely presented, its Dehn function is bounded by a polynomial (hence its word problem is in NP) and its isodiametric function is linear. A version of that result for higher homotopy groups was proved by Riley [25]. The converse statement does not hold: there are finitely presented groups with non-simply connected asymptotic cones and polynomial Dehn functions [1], [26], and even with polynomial Dehn functions and linear isodiametric functions [21]. A partial converse statement was proved by Papasoglu [23]: a group with quadratic Dehn function has all asymptotic cones simply connected (for groups with subquadratic Dehn functions, i.e. -
Zuoqin Wang Time: March 25, 2021 the QUOTIENT TOPOLOGY 1. The
Topology (H) Lecture 6 Lecturer: Zuoqin Wang Time: March 25, 2021 THE QUOTIENT TOPOLOGY 1. The quotient topology { The quotient topology. Last time we introduced several abstract methods to construct topologies on ab- stract spaces (which is widely used in point-set topology and analysis). Today we will introduce another way to construct topological spaces: the quotient topology. In fact the quotient topology is not a brand new method to construct topology. It is merely a simple special case of the co-induced topology that we introduced last time. However, since it is very concrete and \visible", it is widely used in geometry and algebraic topology. Here is the definition: Definition 1.1 (The quotient topology). (1) Let (X; TX ) be a topological space, Y be a set, and p : X ! Y be a surjective map. The co-induced topology on Y induced by the map p is called the quotient topology on Y . In other words, −1 a set V ⊂ Y is open if and only if p (V ) is open in (X; TX ). (2) A continuous surjective map p :(X; TX ) ! (Y; TY ) is called a quotient map, and Y is called the quotient space of X if TY coincides with the quotient topology on Y induced by p. (3) Given a quotient map p, we call p−1(y) the fiber of p over the point y 2 Y . Note: by definition, the composition of two quotient maps is again a quotient map. Here is a typical way to construct quotient maps/quotient topology: Start with a topological space (X; TX ), and define an equivalent relation ∼ on X. -
Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension
LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K. -
The Double Suspension of the Mazur Homology 3-Sphere
THE DOUBLE SUSPENSION OF THE MAZUR HOMOLOGY SPHERE FADI MEZHER 1 Introduction The main objects of this text are homology spheres, which are defined below. Definition 1.1. A manifold M of dimension n is called a homology n-sphere if it has the same homology groups as Sn; that is, ( Z if k ∈ {0, n} Hk(M) = 0 otherwise A result of J.W. Cannon in [Can79] establishes the following theorem Theorem 1.2 (Double Suspension Theorem). The double suspension of any homology n-sphere is homeomorphic to Sn+2. This, however, is beyond the scope of this text. We will, instead, construct a homology sphere, the Mazur homology 3-sphere, and show the double suspension theorem for this particular manifold. However, before beginning with the proper content, let us study the following famous example of a nontrivial homology sphere. Example 1.3. Let I be the group of (orientation preserving) symmetries of the icosahedron, which we recall is a regular polyhedron with twenty faces, twelve vertices, and thirty edges. This group, called the icosahedral group, is finite, with sixty elements, and is naturally a subgroup of SO(3). It is a well-known fact that we have a 2-fold covering ξ : SU(2) → SO(3), where ∼ 3 ∼ 3 SU(2) = S , and SO(3) = RP . We then consider the following pullback diagram S0 S0 Ie := ι∗(SU(2)) SU(2) ι∗ξ ξ I SO(3) Then, Ie is also a group, where the multiplication is given by the lift of the map µ ◦ (ι∗ξ × ι∗ξ), where µ is the multiplication in I. -
L22 Adams Spectral Sequence
Lecture 22: Adams Spectral Sequence for HZ=p 4/10/15-4/17/15 1 A brief recall on Ext We review the definition of Ext from homological algebra. Ext is the derived functor of Hom. Let R be an (ordinary) ring. There is a model structure on the category of chain complexes of modules over R where the weak equivalences are maps of chain complexes which are isomorphisms on homology. Given a short exact sequence 0 ! M1 ! M2 ! M3 ! 0 of R-modules, applying Hom(P; −) produces a left exact sequence 0 ! Hom(P; M1) ! Hom(P; M2) ! Hom(P; M3): A module P is projective if this sequence is always exact. Equivalently, P is projective if and only if if it is a retract of a free module. Let M∗ be a chain complex of R-modules. A projective resolution is a chain complex P∗ of projec- tive modules and a map P∗ ! M∗ which is an isomorphism on homology. This is a cofibrant replacement, i.e. a factorization of 0 ! M∗ as a cofibration fol- lowed by a trivial fibration is 0 ! P∗ ! M∗. There is a functor Hom that takes two chain complexes M and N and produces a chain complex Hom(M∗;N∗) de- Q fined as follows. The degree n module, are the ∗ Hom(M∗;N∗+n). To give the differential, we should fix conventions about degrees. Say that our differentials have degree −1. Then there are two maps Y Y Hom(M∗;N∗+n) ! Hom(M∗;N∗+n−1): ∗ ∗ Q Q The first takes fn to dN fn. -
Arxiv:1812.07604V1 [Math.AT] 18 Dec 2018 Ons Hsvledosa H Ieo H Oe of Model the of Size the As Drops Value This Points
THE TOPOLOGICAL COMPLEXITY OF FINITE MODELS OF SPHERES SHELLEY KANDOLA Abstract. In [2], Farber defined topological complexity (TC) to be the mini- mal number of continuous motion planning rules required to navigate between any two points in a topological space. Papers by [4] and [3] define notions of topological complexity for simplicial complexes. In [9], Tanaka defines a notion of topological complexity, called combinatorial complexity, for finite topologi- cal spaces. As is common with papers discussing topological complexity, each includes a computation of the TC of some sort of circle. In this paper, we compare the TC of models of S1 across each definition, exhibiting some of the nuances of TC that become apparent in the finite setting. In particular, we show that the TC of finite models of S1 can be 3 or 4 and that the TC of the minimal finite model of any n-sphere is equal to 4. Furthermore, we exhibit spaces weakly homotopy equivalent to a wedge of circles with arbitrarily high TC. 1. Introduction Farber introduced the notion of topological complexity in [2] as it relates to motion planning in robotics. Informally, the topological complexity of a robot’s space of configurations represents the minimal number of continuous motion plan- ning rules required to instruct that robot to move from one position into another position. Although topological complexity was originally defined for robots with a smooth, infinite range of motion (e.g. products of spheres or real projective space), it makes sense to consider the topological complexity of finite topological spaces. For example, one could determine the topological complexity of a finite state machine or a robot powered by stepper motors. -
A Suspension Theorem for Continuous Trace C*-Algebras
proceedings of the american mathematical society Volume 120, Number 3, March 1994 A SUSPENSION THEOREM FOR CONTINUOUS TRACE C*-ALGEBRAS MARIUS DADARLAT (Communicated by Jonathan M. Rosenberg) Abstract. Let 3 be a stable continuous trace C*-algebra with spectrum Y . We prove that the natural suspension map S,: [Cq{X), 5§\ -* [Cq{X) ® Co(R), 3 ® Co(R)] is a bijection, provided that both X and Y are locally compact connected spaces whose one-point compactifications have the homo- topy type of a finite CW-complex and X is noncompact. This is used to compute the second homotopy group of 31 in terms of AMheory. That is, [C0(R2), 3] = K0{30) i where 3fa is a maximal ideal of 3 if Y is com- pact, and 3o = 3! if Y is noncompact. 1. Introduction For C*-algebras A, B let Hom(^4, 5) denote the space of (not necessarily unital) *-homomorphisms from A to 5 endowed with the topology of point- wise norm convergence. By definition <p, w e Hom(^, 5) are called homotopic if they lie in the same pathwise connected component of Horn (A, 5). The set of homotopy classes of the *-homomorphisms in Hom(^, 5) is denoted by [A, B]. For a locally compact space X let Cq(X) denote the C*-algebra of complex-valued continuous functions on X vanishing at infinity. The suspen- sion functor S for C*-algebras is defined as follows: it takes a C*-algebra A to A ® C0(R) and q>e Hom(^, 5) to <P0 idem € Hom(,4 ® C0(R), 5 ® C0(R)). -
Arxiv:0704.1009V1 [Math.KT] 8 Apr 2007 Odo References
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES BEHRANG NOOHI These are the notes of three lectures given in the International Workshop on Noncommutative Geometry held in I.P.M., Tehran, Iran, September 11-22. The first lecture is an introduction to the basic notions of abelian category theory, with a view toward their algebraic geometric incarnations (as categories of modules over rings or sheaves of modules over schemes). In the second lecture, we motivate the importance of chain complexes and work out some of their basic properties. The emphasis here is on the notion of cone of a chain map, which will consequently lead to the notion of an exact triangle of chain complexes, a generalization of the cohomology long exact sequence. We then discuss the homotopy category and the derived category of an abelian category, and highlight their main properties. As a way of formalizing the properties of the cone construction, we arrive at the notion of a triangulated category. This is the topic of the third lecture. Af- ter presenting the main examples of triangulated categories (i.e., various homo- topy/derived categories associated to an abelian category), we discuss the prob- lem of constructing abelian categories from a given triangulated category using t-structures. A word on style. In writing these notes, we have tried to follow a lecture style rather than an article style. This means that, we have tried to be very concise, keeping the explanations to a minimum, but not less (hopefully). The reader may find here and there certain remarks written in small fonts; these are meant to be side notes that can be skipped without affecting the flow of the material. -
Regular Cell Complexes in Total Positivity
REGULAR CELL COMPLEXES IN TOTAL POSITIVITY PATRICIA HERSH Abstract. Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over R is a reg- ular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) in- tervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology. 1. Introduction In this paper, the following conjecture of Sergey Fomin and Michael Shapiro from [12] is proven. Conjecture 1.1. Let Y be the link of the identity in the totally non- negative real part of the unipotent radical of a Borel subgroup B in a semisimple, simply connected algebraic group defined and split over R. − − Let Bu = B uB for u in the Weyl group W . Then the stratification of Y into Bruhat cells Y \ Bu is a regular CW decomposition. More- over, for each w 2 W , Yw = [u≤wY \ Bu is a regular CW complex homeomorphic to a ball, as is the link of each of its cells.