DOCSLIB.ORG

Theorem of Van Kampen, Covering Spaces, Examples and Complements

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Theorem of Van Kampen, Covering Spaces, Examples and Complements Notes on Algebraic Topology 1.5 Theorem of Van Kampen The theorem of Van Kampen(28) allows the computation of the fundamental group of a space in terms of the fundamental groups of the open subsets of a suitable cover. Let X be an arcwise connected topological space, (Λ, ) an ordered set, U : λ Λ an ≤ { λ ∈ } open cover of X such that: (a) all Uλ are arcwise connected; (b) λ µ if and only if U U ; ≤ λ ⊆ µ (c) the family U : λ Λ is stable under finite intersections; { λ ∈ } (d) there exists x0 λ Λ Uλ. ∈ ∈ Let us denote by ι :U X and ι : U U the canonical inclusions (where λ λ −→ λ,µ λ −→ µ λ µ), and write for short π (U ) instead of π (U ,x ). We then have morphisms ≤ 1 λ 1 λ 0 ι : π (U ) π (X) and ι : π (U ) π (U )(whereλ µ), which simply λ# 1 λ −→ 1 λ,µ# 1 λ −→ 1 µ ≤ associate to the class of a loop the class of the same loop viewed in the larger space. In particular, one has an inductive system π (U ),ι : λ, µ Λ (see Appendix A.1). { 1 λ λ,µ ∈ } Let M Λsuch that X = λ M Uλ. ⊂ ∈ Proposition 1.5.1. π (X)is generated by ι (π (U )) : λ M . 1 { λ# 1 λ ∈ } 1 Proof. Let γ : I X be a loop at x , δ>0 the Lebesgue number relative to the cover γ− (Uλ):λ M −→ 0 { ∈ } of I,0=t0 <t1 < <tk 1 <tk =1withtj tj 1 <δand λj M (with j =1,...,k)suchthat ··· − − − ∈ γ([tj 1,tj ]) Uλ .Sinceγ(tj ) Uλ Uλ (arcwise connected), let σj (for j =1,...,k 1) be an arc − ⊂ j ∈ j ∩ j+1 − into Uλ Uλ from x to γ(tj ). Setting γj = γ (reparametrized by sending tj 1 to 0 and tj to j j+1 0 [tj 1,tj ] − ∩ 1 | − 1 1 1) it clearly holds [γ]=[γ1] [γk]=[γ1 σ1− ] [σ1 γ2 σ2− ] [σk 2 γk 1 σk− 1][σk 1 γk]. ····· · · · · ····· − · − · − − · An obvious consequence of Proposition 1.5.1 is: Corollary 1.5.2. If the open subsets U (λ M) are simply connected, such is also X. λ ∈ n Example. S is simply connected for n 2. Namely, let N = en+1 be the North pole, S = N the n ≥ n − South pole, and set U = S N and V = S S :notingthatU V is arcwise connected and that \{ } \{ } ∩ both U and V are simply connected, just apply Corollary 1.5.2. Note that this argument does not apply 1 for S (in that case U V is not arcwise connected). ∩ In general one has the following result (see Appendix A.1 for the notion of “inductive limit” of an inductive system). Theorem 1.5.3. (Van Kampen) In the category Groups it holds π1(X)=limπ1(Uλ). λ Λ −→∈ (28)The result has been proved independently also by Karl Seifert in the 30s of last century; in fact, it is often referred to as “Seifert - Van Kampen theorem”. Corrado Marastoni 22 Notes on Algebraic Topology Proof. Consider a group L and a family of morphisms ψλ : π1(Uλ) L such that ψλ = ψµ ιλ,µ# for λ µ, −→ ◦ ≤ and let us see if there exists a unique morphism ψ : π1(X) L such that ψλ = ψ ιλ# for any λ Λ. So let −→ ◦ ∈ [γ] π1(X): by Proposition 1.5.1 we may write [γ]=ιλ #([γ1]) ιλ #([γk]) with [γj ] π1(Uλ ). If ψ ∈ 1 ····· k ∈ j exists, it must be necessarily unique because ψ([γ]) = ψ(ιλ #[γ1]) ψ(ιλ #[γk]) = ψλ ([γ1]) ψλ ([γk]) 1 ··· k 1 ··· k (the products are in L). We are left with showing that this is actually a well-posed definition for ψ,i.e.thatif [c ]=ι ([σ ]) ι ([σ ]) then also ψ ([σ ]) ψ ([σ ]) = e (where e is the identity element of L). x0 λ1 # 1 λk # k λ1 1 λk k ····· j 1 j ··· Let σ = σ1 σk be in X (hence, if t [ −k , k ]itholdsσ(t)=σj (kt (j 1))), and let h : I I X be a ····· ∈ − − 1× −→ homotopy rel ∂I between σ and c .Letε>0 the Lebesgue number relative to the cover h− (U ):λ Λ x0 λ √2 i 1 i j 1{ j ∈ } of I I,andletr N be such that kr <ε.Hence,settingRi,j = k−r , kr k−r , kr I I (for × r ∈ ×i j ⊂ × i, j =1,...,k ), there exists λi,j Λsuchthath(Ri,j ) Uλi,j .Letvi,j = kr , kr (hence Ri,j is the r ∈ ⊂ square with side k− and opposed vertices vi 1,j 1 and vi,j ), Uµ(i,j) the intersection of the (one, two or − − t+(i 1) j four) Uλ such that vi,j Rl,m,andγi,j apathinU from x to h(vi,j ). Let αi,j (t)=h − , l,m ∈ µ(i,j) 0 kr kr i t+(j 1) (path from h(vi 1,j )toh(vi,j )) and βi,j (t)=h r , r− (path from h(vi,j 1)toh(vi,j )): note that − k k − α r 1 α r 1 =[σ ] (for m =1 ,...,k)andthat α r (t)=β (t)=β r (t) x (for (m 1)k − +1,0 mk − ,0 m i,k 0,j k ,j 0 − ····· r ≡ t I and i, j =1,...,k ). From the equality [αi,j 1 βi,j ]=[βi 1,j αi,j ] one gets (by inserting the paths ∈ − · − · 1 1 γi,j and their inverses to base at x ) the relations (γi 1,j 1 αi,j 1) γ− (γi,j 1 βi,j ) γ− = 0 − − − i,j 1 − i,j 1 1 · · − · · · (γi 1,j 1 βi 1,j ) γ− (γi 1,j αi,j ) γ− in the group π1(Uλ ). Applying ψλ and setting − − − i 1,j − i,j i,j i,j · · − · 1 · · 1 ai,j = ψλ (γi 1,j αi,j ) γ− and bi,j = ψµ(i,j) (γi,j 1 βi,j ) γ− ,onethenhastheequality i,j − · · i,j − · · i,j ai,j 1 bi,j = bi 1,j ai,j in L.Knowingthata1,kr akr ,kr = e and that b0,j = bkr ,j = e (for any j = − r − ··· 1,...,k ), one has e = a1,kr akr ,kr =(b0,kr a1,kr )a2,kr akr ,kr = a1,kr 1(b1,kr a2,kr ) akr ,kr = ··· ··· − ··· = a1,kr 1 akr ,kr 1; by repeating the procedure one obtains a1,0 akr ,0 = e, as required. ··· − ··· − ··· Now the problem is to understand what lim π1(Uλ)seemslike. λ Λ −→∈ In general, the free product λ Λ Gλ of a family of groups Gλ : λ Λ is the group ∈ { ∈ } formed by finite “words” a ∗a constructed with “letters” a G (j =1,...,k; k 1), 1 ··· k j ∈ λj ≥ where any letter is different from the identity element in the respective group and where two adjacent letters must belong to different groups (one often says “reduced letters”); also the “empty word” is considered to be an element. The operation is given by the natural juxtaposition (a a ) (b b )=a a b b where, in the case a and 1 ··· k · 1 ··· h 1 ··· k 1 ··· h k b1 belong to a same group, the expression “akb1” should be replaced by their product in that group (and possibly removed if akb1 is the identity, causing then the same procedure for ak 1b2, and so on); the identity element is clearly the empty word. − Example. The free product of any number of copies of Z is called free group, in the sense that there is one generator for each copy of Z and the elements of the group are words formed by powers of these generators. For example, Z Z is formed by the words r1s1 rksk where all rj ’s and sj ’s are integer (the ∗ ··· rj ’s are meant to belong to the first copy of Z,andthesj ’s to the second); or also, in abstract notation, by ak1 bh1 akr bhr where a and b denote the two generators and the exponents are integers. ··· Note that for any µ Λthere is a natural monomorphism Gµ λ Λ Gλ; in fact one sees ∈ −→ ∈ that the free product λ Λ Gλ is the inductive limit in Groups of∗ the system Gλ : λ Λ ∈ { ∈ } with trivial preorder, i.e.∗ without considering morphisms(29). More generally, if morphisms f : G G are given for some pairs (λ, µ)withf f = f whenever defined, λ,µ λ −→ µ λ,µ ◦ µ,ν λ,ν then to obtain the inductive limit of the system G ,f : λ, µ Λ one must quotient { λ λ,µ ∈ } out the previous free product λ Λ Gλ by its normal subgroup N generated by all the elements of type f (a)f (a)∗1 ∈for a G whenever the morphisms f and f are λ,µ λ,ν − ∈ λ λ,µ λ,ν (29) Namely, given a family of morphisms ψλ : Gλ L, the definition (necessary, hence unique) −→ ψ(a1 ak)=ψλ (a1) ψλ (ak) is a morphism. ··· 1 ··· k Corrado Marastoni 23 Notes on Algebraic Topology defined(30): this procedure is said to be an amalgamation of the free product with respect to the given morphisms fλ,µ. When applying the above notions to the framework of Van Kampen, the groups are π1(Uλ) and the morphisms are the maps ι : π (U ) π (U ), which send the class of a loop λ,µ# 1 λ −→ 1 µ in the small open subset Uλ to the class of same loop viewed in the larger open subset Uµ: by Proposition 1.5.1 and the subsequent discussion it is then enough to consider the free product of the groups π1(Uλ)’s of a selected family of open subsets Uλ with indices λ M which cover X and are not contained in each other, and then to amalgamate only ∈ with respect to the double intersections U U for λ, µ M.(31) λ ∩ µ ∈ Example.
Load more

Recommended publications
  • Arxiv:1712.06224V3 [Math.MG] 27 May 2018 Yusu Wang3 Department of Computer Science, the Ohio State University
    Vietoris–Rips and Čech Complexes of Metric Gluings 03:1 Vietoris–Rips and Čech Complexes of Metric Gluings Michał Adamaszek MOSEK ApS Copenhagen, Denmark [email protected] https://orcid.org/0000-0003-3551-192X Henry Adams Department of Mathematics, Colorado State University Fort Collins, CO, USA [email protected] https://orcid.org/0000-0003-0914-6316 Ellen Gasparovic Department of Mathematics, Union College Schenectady, NY, USA [email protected] https://orcid.org/0000-0003-3775-9785 Maria Gommel Department of Mathematics, University of Iowa Iowa City, IA, USA [email protected] https://orcid.org/0000-0003-2714-9326 Emilie Purvine Computing and Analytics Division, Pacific Northwest National Laboratory Seattle, WA, USA [email protected] https://orcid.org/0000-0003-2069-5594 Radmila Sazdanovic1 Department of Mathematics, North Carolina State University Raleigh, NC, USA [email protected] https://orcid.org/0000-0003-1321-1651 Bei Wang2 School of Computing, University of Utah Salt Lake City, UT, USA [email protected] arXiv:1712.06224v3 [math.MG] 27 May 2018 https://orcid.org/0000-0002-9240-0700 Yusu Wang3 Department of Computer Science, The Ohio State University 1 Simons Collaboration Grant 318086 2 NSF IIS-1513616 and NSF ABI-1661375 3 NSF CCF-1526513, CCF-1618247, CCF-1740761, and DMS-1547357 Columbus, OH, USA [email protected] https://orcid.org/0000-0001-7950-4348 Lori Ziegelmeier Department of Mathematics, Statistics, and Computer Science, Macalester College Saint Paul, MN, USA [email protected] https://orcid.org/0000-0002-1544-4937 Abstract We study Vietoris–Rips and Čech complexes of metric wedge sums and metric gluings.
    [Show full text]
  • ALGEBRAIC TOPOLOGY Contents 1. Preliminaries 1 2. the Fundamental
    ALGEBRAIC TOPOLOGY RAPHAEL HO Abstract. The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology. Contents 1. Preliminaries 1 2. The Fundamental Group 2 3. Van Kampen's Theorem 5 4. Covering Spaces 6 5. Graphs 9 Acknowledgements 11 References 11 1. Preliminaries Notations 1.1. I [0, 1] the unit interval Sn the unit sphere in Rn+1 × standard cartesian product ≈ isomorphic to _ the wedge sum A − B the space fx 2 Ajx2 = Bg A=B the quotient space of A by B. In this paper we assume basic knowledge of set theory. We also assume previous knowledge of standard group theory, including the notions of homomorphisms and quotient groups. Let us begin with a few reminders from algebra. Definition 1.2. A group G is a set combined with a binary operator ? satisfying: • For all a; b 2 G, a ? b 2 G. • For all a; b; c 2 G,(a ? b) ? c = a ? (b ? c). • There exists an identity element e 2 G such that e ? a = a ? e = a. • For all a 2 G, there exists an inverse element a−1 2 G such that a?a−1 = e. A convenient way to describe a particular group is to use a presentation, which consists of a set S of generators such that each element of the group can be written Date: DEADLINE AUGUST 22, 2008. 1 2 RAPHAEL HO as a product of elements in S, and a set R of relations which define under which conditions we are able to simplify our `word' of product of elements in S.
    [Show full text]
  • Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
    MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X).
    [Show full text]
  • 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.
    [Show full text]
  • The Capacity of Wedge Sum of Spheres of Different Dimensions
    The Capacity of Wedge Sum of Spheres of Different Dimensions Mojtaba Mohareri, Behrooz Mashayekhy∗, Hanieh Mirebrahimi Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran. Abstract K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and raised some interesting questions about it. In this paper, during computing the capacity of wedge sum of finitely many spheres of different dimensions and the complex projective plane, we give a negative answer to a question of Borsuk whether the capacity of a compactum determined by its homology properties. Keywords: Homotopy domination, Homotopy type, Moore space, Polyhedron, CW-complex, Compactum. 2010 MSC: 55P15, 55P55, 55P20,54E30, 55Q20. 1. Introduction and Motivation K. Borsuk in [2], introduced the concept of the capacity of a compactum (compact metric space) as follows: the capacity C(A) of a compactum A is the cardinality of the set of all shapes of compacta X for which Sh(X) 6 Sh(A) (for more details, see [10]). For polyhedra, the notions shape and shape domination in the above definition arXiv:1611.00487v1 [math.AT] 2 Nov 2016 can be replaced by the notions homotopy type and homotopy domination, respec- tively. Indeed, by some known results in shape theory one can conclude that for any polyhedron P , there is a one to one functorial correspondence between the shapes of compacta shape dominated by P and the homotopy types of CW-complexes (not ∗Corresponding author Email addresses: [email protected] (Mojtaba Mohareri), [email protected] (Behrooz Mashayekhy), h−[email protected] (Hanieh Mirebrahimi) Preprint submitted to September 20, 2018 necessarily finite) homotopy dominated by P (for both pointed and unpointed poly- hedra) [6].
    [Show full text]
  • (Jerry) Luo As Its Name May Suggest, the Fundamental Group of a Space Is One of the Most Important Ideas from Algebraic Topology
    FINAL MATH 232A Name: Jiajie (Jerry) Luo As its name may suggest, the fundamental group of a space is one of the most important ideas from algebraic topology. It's an algebraic invariant for topological spaces describing the (closed) loops up to homotopy in that space. Much can be learned about topological spaces through the fundamental group, as well as its generalizations - the higher homotopy groups. While homotopy groups tell us quite a lot about a space, they're limited in accessibility, as computing them have been a challenge. This motivates the construction of another algebraic invariant for topological spaces - homology groups. Unlike, homotopy groups, homology groups are much more accessible, as they are easier to compute. The drawback in this is that they are less intuitive and more of a challenge to define. However, despite this, we will show the power that homology theory possesses in extracting information about topological spaces. We begin by discussing simplicial homology, which is much nicer and more intuitive than its more advanced counterpart, singular homology. To do this, we first introduce the structure on which simplicial homology is built upon - ∆-complex structures. The construction for ∆-complex structures, in a more or less way, is representing a topological space with (generalized) triangles. We define an n-simplex as the convex hull of n + 1 points, m fv0; ··· ; vng, that sit in R , where fv0; ··· ; vng do not lie in a hyperplane of dimension less than n (note that this requires m ≥ n). A simple example is taking the convex hull of the standard n+1 (orthonormal) basis elements on R .
    [Show full text]
  • Topology I Humboldt-Universität Zu Berlin C. Wendl / F. Schmäschke
    Topology I Humboldt-Universitat¨ zu Berlin C. Wendl / F. Schmaschke¨ Summer Semester 2017 PROBLEM SET 7 Due: 7.06.2017 Instructions Problems marked with (∗) will be graded. Solutions may be written up in German or English and should be handed in before the Ubung¨ on the due date. For problems without (∗), you do not need to write up your solutions, but it is highly recommended that you think through them before the next Wednesday lecture. 1 Special note: You may continue to use the fact that π1(S ) =∼ Z without proof. (We’ll prove it next week.) Problems 1. Recall that the wedge sum of two pointed spaces (X, x) and (Y,y) is defined as X ∨ Y = (X ⊔ Y )/∼ where the equivalence relation identifies the two base points x and y. It is commonly said that whenever X and Y are both path-connected and are otherwise “reasonable” spaces, the formula π1(X ∨ Y ) =∼ π1(X) ∗ π1(Y ) (1) holds. We’ve seen for instance that this is true when X and Y are both circles. The goal of this problem is to understand slightly better what “reasonable” means in this context, and why such a condition is needed. (a) Show by a direct argument (i.e. without trying to use Seifert-van Kampen) that if X and Y are both Hausdorff and simply connected, then X ∨ Y is simply connected. Hint: Hausdorff implies that X \{x} and Y \{y} are both open subsets. Consider loops γ : [0, 1] → X ∨ Y based at [x] = [y] and decompose [0, 1] into subintervals in which γ(t) stays in either X or Y .
    [Show full text]
  • Example Sheet 7
    MA3F1 Introduction to Topology November 11, 2019 Example Sheet 7 Please let me (Martin) know if any of the problems are unclear or have typos. Please turn in a solution to one of Exercises (7.1) or (7.4) by 12:00 on 21/11/2019, to the dropoff box in front of the undergraduate office. If you collaborate with other students, please include their names. For the next three problems we need the following definition. Suppose X is a topological space. We define CX to be the cone on X: that is, CX = X × I=(x; 1) ∼ (y; 1) for all x; y 2 X. The point a = [(x; 1)] is called the apex of the cone. (7.1) Equip the integers Z with the discrete topology. Show that CZ is homeomorphic to the wedge sum of a countable collection of unit intervals at 1. Solution (7.1) The disjoint union of a countable collection of unit intervals is defined to be S fng × I = × I, the latter with the product topology. The wedge sum n2Z Z _n2ZI at 1 is then defined in the same way as the cone CZ. 2 (7.2) Let In ⊂ R to be the line segment connecting (0; 1) to (n; 0), for n 2 Z. Set D = [n2ZIn and equip D with the subspace topology. Show that CZ is not homeomorphic to D. Solution (7.2) Suppose that they are homeomorphic via q : D ! CZ. Note that q sends the apex to the apex (removing the apex gives a space with a countable number of connected components in both cases, and the apex is the only point with this property).
    [Show full text]
  • Notes on Category Theory (In Progress)
    Notes on Category Theory (in progress) George Torres Last updated February 28, 2018 Contents 1 Introduction and Motivation 3 2 Definition of a Category 3 2.1 Examples . .4 3 Functors 4 3.1 Natural Transformations . .5 3.2 Adjoint Functors . .5 3.3 Units and Counits . .6 3.4 Initial and Terminal Objects . .7 3.4.1 Comma Categories . .7 4 Representability and Yoneda's Lemma 8 4.1 Representables . .9 4.2 The Yoneda Embedding . 10 4.3 The Yoneda Lemma . 10 4.4 Consequences of Yoneda . 11 5 Limits and Colimits 12 5.1 (Co)Products, (Co)Equalizers, Pullbacks and Pushouts . 13 5.2 Topological limits . 15 5.3 Existence of limits and colimits . 15 5.4 Limits as Representable Objects . 16 5.5 Limits as Adjoints . 16 5.6 Preserving Limits and GAFT . 18 6 Abelian Categories 19 6.1 Homology . 20 6.1.1 Biproducts . 21 6.2 Exact Functors . 23 6.3 Injective and Projective Objects . 26 6.3.1 Projective and Injective Modules . 27 6.4 The Chain Complex Category . 28 6.5 Homological dimension . 30 6.6 Derived Functors . 32 1 CONTENTS CONTENTS 7 Triangulated and Derived Categories 35 ||||||||||| Note to the reader: This is an ongoing collection of notes on introductory category theory that I have kept since my undergraduate years. They are aimed at students with an undergraduate level background in topology and algebra. These notes started as lecture notes for the Fall 2015 Category Theory tutorial led by Danny Shi at Harvard. There is no single textbook that these notes follow, but Categories for the Working Mathematician by Mac Lane and Lang's Algebra are good standard resources.
    [Show full text]
  • 1) Let X Be a Topological Space, and a ⊂ X Be a Subspace. a Retraction of X Onto a Is a Map R : X → X Such That R(X) = a and R|A = Id
    Homework 1: 1) Let X be a topological space, and A ⊂ X be a subspace. A retraction of X onto A is a map r : X ! X such that r(X) = A and rjA = id. Show that r2 = r. Show that if X is contractible and r : X ! X is a retraction of X onto A, then A is also contractible. 2) A deformation retraction of a space X onto a subspace A is a family of maps ft : X ! X, t 2 I, such that f0 = id, f1(X) = A and ftjA = id for all t. As usual, the family ft should be continuous, that is, the associated map X×I ! X is continuous. Construct a deformation retraction of Rn+1nf0g ! Sn. 3) The \dunce cap" is the quotient of a triangle (and interior) obtained by identifying all the three edges in an inconsistent manner. That is, if the vertices of the triangle are p; q; r then we identify the line segment (p; q) with (q; r) and with (p; r) in the orientation indicated by the order given. Show that the dunce cap is contractible. 4) Let fXαgα2I be a collection of spaces indexed by a set I. Let pα 2 Xα be W basepoints. Define the wedge sum α Xα as the quotient space of the disjoint union of Xα by the equivalence relation pα ∼ pβ for α; β 2 I. Show that, for n ≥ 1, the complement of m distinct points in Rn is homotopy equivalent to the wedge sum of m copies of the sphere Sn−1 = fx 2 Rn : jxj = 1g.
    [Show full text]
  • Mayer-Vietoris Sequence and CW Complexes
    Homework 4: Mayer-Vietoris Sequence and CW complexes Due date: Friday, October 4th. 0. Goals and Prerequisites The goal of this homework assignment is to begin using the Mayer-Vietoris sequence and cellular homology to compute homology of basic shapes. Some proofs depend on algebraic manipulations, while others depend on geometric arguments. So this is a good first run at trying to do problems that are genuinely problems of algebraic topology. You may find many of these problems difficult, as some will require some geometric creativity. Don’t be discouraged, in particular, if the problems about surfaces seem difficult. It takes some time and wisdom to develop the intuition for CW structures. You need to know what CW homology is, and how to use the Mayer-Vietoris sequence. This can be found in the class notes. 1. Reduced homology Recall that reduced homology H˜n(X) for a non-empty space X is defined to be ˜ ˜ Hn(X) ∼= Hn(X), H0(X) ∼= ker(/ image(∂1)) where is the map of abelian groups C0(X) Z, aixi ai. → → If X is non-empty and x X is a basepoint, show 0 ∈ ˜ H0(X) ∼= H0(X, x0) where the latter is relative homology. 2. Homology of wedges Let X and Y be spaces, and choose a point x0 X, y0 Y in each space. The wedge sum of the two spaces is defined to be the space ∈ ∈ X Y := X Y = (X Y )/x y . ∨ ∼ 0 ∼ 0 x y 0∼ 0 This is the space obtained by gluing x0 to y0. For instance, if X and Y are both homeomorphic to S1, their wedge sum is a figure 8.
    [Show full text]
  • Wise 20/21 Algebraic Topology I Exercise Sheet 8 (Due January 13)
    WiSe 20/21 Algebraic Topology I Exercise sheet 8 (due January 13) Exercise 8.1. Consider a commutative diagram of abelian groups fn gn @n ··· An Bn Cn An−1 ··· an bn =∼ cn an−1 0 0 0 0 fn 0 gn 0 @n 0 ··· An Bn Cn An−1 ··· where the rows are long exact sequences and cn is an isomorphism for all n 2 Z. Verify that the associated \abstract Mayer{Vietoris" sequence 0 MV (an;−fn) 0 (fn;bn) 0 @n ··· An An ⊕ Bn Bn An−1 ··· MV −1 0 is exact, where @n = @n ◦ cn ◦ gn. Exercise 8.2. Let C and D be small categories, f; g : C ! D functors, and φ: f ) g a natural transformation. (a) Show that φ induces a homotopy between the morphisms of simplicial sets N(f);N(g): N(C) ! N(D) and of spaces jN(f)j; jN(g)j: jN(C)j ! jN(D)j: Hint. The natural transformation φ can be viewed as a functor C × [1] ! D. (b) Deduce that jN(−)j takes equivalences of categories to homotopy equivalences of spaces. (c) Suppose that C has either an initial object or a final object. Show that jN(C)j is contractible. Exercise 8.3. Let (Xα)α2A be a family of topological spaces with base points xα 2 Xα. Recall that their coproduct in Top∗ is given by the wedge sum ! _ a Xα = Xα =fxαgα2A: α2A α2A W W Let iβ : Xβ ,! α2A Xα be the canonical map and let pβ : α2A Xα Xβ be the pointed map such that pβ ◦ iα is the identity on Xβ if α = β and is constant otherwise.
    [Show full text]