DOCSLIB.ORG

CONFIGURATION SPACES in ALGEBRAIC TOPOLOGY: LECTURE 2 We Begin Our Study of Configuration Spaces by Observing a Few of Their

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CONFIGURATION SPACES in ALGEBRAIC TOPOLOGY: LECTURE 2 We Begin Our Study of Configuration Spaces by Observing a Few of Their CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X ! Y is an injective continuous map, we have the Σk-equivariant factorization k k f / k XO YO Confk(f) Confk(X) / Confk(Y ) and thus an induced map Bk(f): Bk(X) ! Bk(Y ). This construction respects composition and identities by inspection, so we have functors inj Σk Confk : Top ! Top inj Bk : Top ! Top; where Topinj denotes the category of topological spaces and injective continuous maps, TopΣk the category of Σk-spaces and equivariant maps, and Top the category of topological spaces. Recall that a continuous map is said to be an open embedding if it is both injective and open. Proposition. If f : X ! Y is an open embedding, then Confk(f) : Confk(X) ! Confk(Y ) and Bk(f): Bk(X) ! Bk(Y ) are also open embeddings. Proof. From the definition of the product topology, f k : Xk ! Y k is an open embedding, and the first claim follows. The second claim follows from the first and the fact that π : Confk(X) ! Bk(X) is a quotient map. This functor also respects a certain class of weak equivalences. Definition. An injective continuous map f : X ! Y is an isotopy equivalence if there is an 0 injective continuous map g : Y ! X and homotopies H : g ◦ f =) idX and H : f ◦ g =) idY 0 such that Ht and Ht are both injective for each t 2 [0; 1]. Proposition. If f : X ! Y is an isotopy equivalence, then Confk(f) : Confk(X) ! Confk(Y ) is a homotopy equivalence. Date: 1 September 2017. 1 2 BEN KNUDSEN Proof. In the solid commuting diagram k Xk × [0; 1]k ∼ (X × [0; 1])k H / Xk O 6 O H∆ idXk ×∆k Xk × [0; 1] O Confk(X) × [0; 1] / Confk(X); the diagonal composite is given by the formula ∆ Ht (x1; : : : ; xk) = (Ht(x1);:::;Ht(xk)): By assumption, Ht is injective for each t 2 [0; 1], so the dashed filler exists. By construction, this map restricts to Confk(g ◦ f) = Confk(g) ◦ Confk(f) at t = 0 and to idConfk(X) at t = 1. 0 Applying the same argument to Ht completes the proof. Corollary. If f : X ! Y is an isotopy equivalence then Bk(f): Bk(X) ! Bk(Y ) is a homotopy equivalence. Proof. The homotopies constructed in the proof of the previous corollary are homotopies through Σk-equivariant maps, so they descend to the unordered configuration space. Remark. Another point of view on the previous two results is provided by the fact (which we will not prove here) that Confk and Bk are enriched functors, where the space of injective continuous maps from X to Y is given the subspace topology induced by the compact-open topology on Map(X; Y ). Taking this fact for granted, these results follows immediately, since a homotopy through injective maps is simply a path in this mapping space. Example. If M is a manifold with boundary, then @M admits a collar neighborhood U =∼ t @M × (0; 1]. We define a map r : M ! M by setting rj@M×(0;1](x; t) = (x; 2 ) and extending by the identity. This map is injective, and dilation defines homotopies through injective maps from r ◦ i : M˚ ! M˚ and i ◦ r : M ! M to the respective identity maps. It follows that the induced map Confk(M˚) ! Confk(M) is a homotopy equivalence. These functors also interact well with the operation of disjoint union. Proposition. Let X and Y be topological spaces. The natural map a Confi(X) × Confj(Y ) ×Σi×Σj Σk ! Confk(X q Y ) i+j=k is a Σk-equivariant homeomorphism. In particular, the natural map a Bi(X) × Bj(Y ) ! Bk(X × Y ) i+j=k is a homeomorphism. CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 3 Proof. From the definitions, the dashed filler exists in the commuting diagram a ' Xi × Y j × Σ / (X q Y )k Σi×Σj k O i+j=k O a / Confi(X) × Confj(Y ) ×Σi×Σj Σk Confk(X q Y ) i+j=k and is easily seen to be a bijection, which implies the first claim, since the vertical arrows are inclusions of subspaces. The second claim follows from the first after taking the quotient by the action of Σk. Thus, we may restrict attention to connected background spaces whenever it is convenient to do so. Our next goal is to come to grips with the local structure of configuration spaces. We assume from now on that X is locally path connected, and we fix a basis B for the topology of the space X consisting of connected subsets. We define two partially ordered sets as follows. ∼ k (1) We write Bk = fU ⊆ X : U = qi=1Ui;Ui 2 Bg, and we impose the order relation U ≤ V () U ⊆ V and π0(U ⊆ V ) is surjective: Σ ' (2) We write Bk = f(U; σ): U 2 Bk; σ : f1; : : : ; kg −! π0(U)g, and we impose the order relation (U; σ) ≤ (V; τ) () U ≤ V and τ = σ ◦ π0(U ⊆ V ): Σ Denoting the poset of open subsets of a space Y by O(Y ), there is an inclusion Bk ! O(Confk(X)) of posets defined by 0 U 7! Confk(U; σ) := (x1 : : : ; xk) 2 Confk(U): xi 2 Uσ(i) ⊆ Confk(X) and similarly an inclusion Bk ! O(Bk(X)) defined by 0 U 7! Bk(U) := ffx1; : : : ; xkg 2 Bk(U): fx1; : : : ; xkg \ Ui 6= ?; 1 ≤ i ≤ kg ⊆ Bk(X): Note that these subsets are in fact open, since U ⊆ X is open and configuration spaces respect open embeddings. ' Lemma. For any U 2 Bk and σ : f1; : : : ; kg −! π0(U), there are canonical homeomorphisms k 0 ∼ 0 ∼ Y Bk(U) = Confk(U; σ) = Uσ(i): i=1 Proof. It is easy to see from the definitions that the dashed fillers in the commuting diagram Xk o Conf (X) / B (X) O kO k O k Y 0 0 Uσ(i) o Confk(U; σ) / Bk(U) i=1 exist and are bijections. Since the lefthand map is the inclusion of a subspace and the righthand map is a quotient map, the claim follows. Proposition. Let X be a locally path connected Hausdorff space and B a topological basis for X consisting of connected subsets. 4 BEN KNUDSEN 0 Σ (1) The collection fConfk(U; σ):(U; σ) 2 Bk g ⊆ O(Confk(X)) is a topological basis. 0 (2) The collection fBk(U): U 2 Bkg ⊆ O(Bk(X)) is a topological basis. k Proof. Since X is Hausdorff, Confk(X) is open in X ; therefore, by the definition of the product and subspace topologies, it will suffice for the first claim to show that, given (x1; : : : ; xk) 2 V ⊆ Xk such that ∼ Qk • V = i=1 Vi for open subsets xi 2 Vi ⊆ X, and • V ⊆ Confk(X), Σ 0 there exists (U; σ) 2 Bk with (x1; : : : ; xk) 2 Confk(U; σ) ⊆ V . Now, since B is a topological basis, we may find Ui 2 B with xi 2 Ui ⊆ Vi. The second condition implies that the Vi are `k pairwise disjoint, so we may set U = i=1 Ui and take σ(i) = [Ui]. With these choices k k 0 ∼ Y Y (x1; : : : ; xk) 2 Confk(U; σ) = Ui ⊆ Vi = V; i=1 i=1 as desired. The second claim follows from first, the fact that π : Confk(X) ! Bk(X) is a quotient map, 0 0 and the fact that π(Confk(U; σ)) = Bk(U) for every σ. These and related bases will be important for our later study, when we come to hypercover methods. For now, we draw the following consequences. Corollary. Let X be a locally path connected Hausdorff space. The projection π : Confk(X) ! Bk(X) is a covering space. Proof. For U 2 Bk, we have Σk-equivariant identifications −1 0 [ 0 ∼ 0 π (Bk(U)) = Confk(U; σ) = Bk(U) × Σk; σ:f1;:::;kg∼=π0(U) where the second is induced by a choice of ordering of π0(U). Corollary. If M is an n-manifold, then Confk(M) and Bk(M) are nk-manifolds. Proof. We take B to be the set of Euclidean neighborhoods in M, in which case 0 ∼ 0 ∼ nk Bk(U) = Confk = R ' for any U 2 Bk and σ : f1; : : : ; kg −! π0(U). Exercise. When is Bk(M) orientable? From now on, unless specified otherwise, we take our background space to be a manifold M. In this case, we have access to a poweful tool relating configuration spaces of different cardinalities. The starting point is the observation that the natural projections from the product factor through the configuration spaces as in the following commuting diagram: ` Conf`(M) / M (x1; : : : ; x`) _ k Confk(M) / M (x1; : : : ; xk) (we take the projection to be on the last ` − k coordinates for simplicity, but it is not necessary to make this restriction). Clearly, the fiber over a configuration (x1; : : : ; xk) in the base is the configuration space Conf`−k(M n fx1; : : : ; xkg). Our first theorem asserts that the situation is in fact much better than this. CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 5 Recollection. Recall that, if f : X ! Y is a continuous map, the mapping path space of f is the space of paths in Y out of the image of f. In other words, it is the pullback in the diagram [0;1] Ef / Y p _ f X / Y p(0): The inclusion X ! Ef given by the constant paths is a homotopy equivalence, and evaluation at 1 defines a map πf : Ef ! Y , which is a fibration [May99, 7.3].
Load more

Recommended publications
  • 1.1.5 Hausdorff Spaces
    1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x).
    [Show full text]
  • The Non-Hausdorff Number of a Topological Space
    THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum.
    [Show full text]
  • MTH 304: General Topology Semester 2, 2017-2018
    MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds..........................
    [Show full text]
  • General Topology
    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).
    [Show full text]
  • DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
    DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V .
    [Show full text]
  • 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.
    [Show full text]
  • FINITE TOPOLOGICAL SPACES 1. Introduction: Finite Spaces And
    FINITE TOPOLOGICAL SPACES NOTES FOR REU BY J.P. MAY 1. Introduction: finite spaces and partial orders Standard saying: One picture is worth a thousand words. In mathematics: One good definition is worth a thousand calculations. But, to quote a slogan from a T-shirt worn by one of my students: Calculation is the way to the truth. The intuitive notion of a set in which there is a prescribed description of nearness of points is obvious. Formulating the “right” general abstract notion of what a “topology” on a set should be is not. Distance functions lead to metric spaces, which is how we usually think of spaces. Hausdorff came up with a much more abstract and general notion that is now universally accepted. Definition 1.1. A topology on a set X consists of a set U of subsets of X, called the “open sets of X in the topology U ”, with the following properties. (i) ∅ and X are in U . (ii) A finite intersection of sets in U is in U . (iii) An arbitrary union of sets in U is in U . A complement of an open set is called a closed set. The closed sets include ∅ and X and are closed under finite unions and arbitrary intersections. It is very often interesting to see what happens when one takes a standard definition and tweaks it a bit. The following tweaking of the notion of a topology is due to Alexandroff [1], except that he used a different name for the notion. Definition 1.2. A topological space is an A-space if the set U is closed under arbitrary intersections.
    [Show full text]
  • Is Being Hausdorff a Homotopy Invariant?
    Topology Class Is being Hausdor a homotopy invariant? Yash Chandra Published on: Oct 08, 2020 License: Creative Commons Attribution 4.0 International License (CC-BY 4.0) Topology Class Is being Hausdor a homotopy invariant? Hi Everyone, In class, the concepts of homotopy and homotopy equivalence were defined. In that discussion, homotopy invariants – properties of a topological space that remain invariant under an homotopy equivalence – were introduced. Prof. Terilla left this as an open-ended question whether being Hausdorff is a homotopy invariant or not. We will try to address that question here. I’d like to thank Prof. Terilla for his valuable insights into this. I’d also like to acknowledge fellow Graduate Center students Ben, James, Moshe, and Ryan for informal discussions. The answer to the question is No, being Hausdorff is not a homotopy invariant! We will give a particular example to show that, and then try to give a generic recipe for getting more examples. Before doing anything technical, I think it is reasonable to guess that being Hausdorff is not a homotopy invariant, since homotopy is basically a topological deformation and one would intuitively expect that a non-Hausdorff space can be deformed into a Hausdorff one and vice-versa. Notice that we know that the one-point space is vacuously Hausdorff, so maybe we can find a homotopy equivalence between a non-Hausdorff space and the one-point space. We will try to exploit this idea here. A particular example: Consider the set X = {a, b} with the topology τ = {∅, {a}, X}. (X, τ ) is also known as the Sierpinski Space.
    [Show full text]
  • Math 871-872 Qualifying Exam May 2013 Do Three Questions from Section a and Three Questions from Section B
    Math 871-872 Qualifying Exam May 2013 Do three questions from Section A and three questions from Section B. You may work on as many questions as you wish, but indicate which ones you want graded. When in doubt about the wording of a problem or what results may be assumed without proof, ask for clarification. Do not interpret a problem in such a way that it becomes trivial. Section A: Do THREE problems from this section (1) Let Xα be non-empty topological spaces and suppose that X = QαXα is endowed with the product topology. (a) Prove that each projection map πα is continuous and open. (b) Prove that X is Hausdorff if and only if each space Xα is Hausdorff. (2) (a) Prove that every compact subspace of a Hausdorff space is closed. Show by example that the Hausdorff hypothesis cannot be removed. (b) Prove that every compact Hausdorff space is normal. 2 2 2 2 2 (3) Define an equivalence relation ∼ on R by (x1,y1) ∼ (x2,y2) if and only if x1 + y1 = x2 + y2. (a) Identify the quotient space X = R2/∼ as a familiar space and prove that it is homeo- morphic to this familiar space. (b) Determine whether the natural map p : R2 → X is a covering map. Justify your answer. (4) A space is locally connected if for each point x ∈ X and every neighborhood U of x, there is a connected neighborhood V of x contained in U. (a) Prove that X is locally connected if and only if for every open set U of X, each connected component of U is open in X.
    [Show full text]
  • 9. Stronger Separation Axioms
    9. Stronger separation axioms 1 Motivation While studying sequence convergence, we isolated three properties of topological spaces that are called separation axioms or T -axioms. These were called T0 (or Kolmogorov), T1 (or Fre´chet), and T2 (or Hausdorff). To remind you, here are their definitions: Definition 1.1. A topological space (X; T ) is said to be T0 (or much less commonly said to be a Kolmogorov space), if for any pair of distinct points x; y 2 X there is an open set U that contains one of them and not the other. Recall that this property is not very useful. Every space we study in any depth, with the exception of indiscrete spaces, is T0. Definition 1.2. A topological space (X; T ) is said to be T1 if for any pair of distinct points x; y 2 X, there exist open sets U and V such that U contains x but not y, and V contains y but not x. Recall that an equivalent definition of a T1 space is one in which all singletons are closed. In a space with this property, constant sequences converge only to their constant values (which need not be true in a space that is not T1|this property is actually equivalent to being T1). Definition 1.3. A topological space (X; T ) is said to be T2, or more commonly said to be a Hausdorff space, if for every pair of distinct points x; y 2 X, there exist disjoint open sets U and V such that x 2 U and y 2 V .
    [Show full text]
  • On the Homotopy Groups of Separable Metric Spaces ∗ F.H
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 1607–1614 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On the homotopy groups of separable metric spaces ∗ F.H. Ghane , Hadi Passandideh, Z. Hamed Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran article info abstract Article history: The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. Received 2 June 2010 First, we state a nerve theorem. It gives sufficient conditions under which there is a weak Received in revised form 12 May 2011 n-equivalence between the nerve of a good cover and its underlying space. Then we con- Accepted 19 May 2011 clude that for any (n − 1)-connected, locally (n − 1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, (X),is MSC: πn 55Q05 finitely presented. This result allows us to provide a new proof for a generalization of 55Q52 Shelah’s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 54E35 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topo- 57N15 logical space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic Keywords: 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite Nerve which introduce here.
    [Show full text]
  • Topology: Notes and Problems
    TOPOLOGY: NOTES AND PROBLEMS Abstract. These are the notes prepared for the course MTH 304 to be offered to undergraduate students at IIT Kanpur. Contents 1. Topology of Metric Spaces 1 2. Topological Spaces 3 3. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. Infinitude of Prime Numbers 6 5. Product Topology 6 6. Subspace Topology 7 7. Closed Sets, Hausdorff Spaces, and Closure of a Set 9 8. Continuous Functions 12 8.1. A Theorem of Volterra Vito 15 9. Homeomorphisms 16 10. Product, Box, and Uniform Topologies 18 11. Compact Spaces 21 12. Quotient Topology 23 13. Connected and Path-connected Spaces 27 14. Compactness Revisited 30 15. Countability Axioms 31 16. Separation Axioms 33 17. Tychonoff's Theorem 36 References 37 1. Topology of Metric Spaces A function d : X × X ! R+ is a metric if for any x; y; z 2 X; (1) d(x; y) = 0 iff x = y. (2) d(x; y) = d(y; x). (3) d(x; y) ≤ d(x; z) + d(z; y): We refer to (X; d) as a metric space. 2 Exercise 1.1 : Give five of your favourite metrics on R : Exercise 1.2 : Show that C[0; 1] is a metric space with metric d1(f; g) := kf − gk1: 1 2 TOPOLOGY: NOTES AND PROBLEMS An open ball in a metric space (X; d) is given by Bd(x; R) := fy 2 X : d(y; x) < Rg: Exercise 1.3 : Let (X; d) be your favourite metric (X; d). How does open ball in (X; d) look like ? Exercise 1.4 : Visualize the open ball B(f; R) in (C[0; 1]; d1); where f is the identity function.
    [Show full text]