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Algebraic Topology Wojciech Kryszewski Algebraic Topology For Practitioners Uniwersytet Mikołaja Kopernika Toruń 2009 Contents Introduction 1 0.1 General notation . 1 0.2 General topology . 1 0.3 Differential Calculus . 6 0.4 Algebra . 9 0.4.A Exterior Algebra . 9 0.4.B Complex exterior algebra . 12 0.4.C Free R-Modules; Specker modules . 15 0.4.D Exact sequences . 16 0.4.E Functors ⊗, Hom and Ext ................................ 17 0.4.F Direct and inverse systems . 19 0.4.G Limits and (co)homology . 30 0.5 Functional and nonsmooth analysis . 31 1 Categories, Functors and Algebraization of Topological Problems 32 1.1 Categories and Functors . 32 1.2 Algebraization . 34 1.3 Theory of homotopy . 37 1.3.A The space of continuous maps . 37 1.3.B Extensions . 39 1.3.C Homotopy . 41 1.3.D Special Homotopies . 45 1.3.E Homotopy Sets . 46 1.3.F The Index of a Function relative to a Curve . 49 1.4 Absolute Homotopy Groups of a Space . 54 2 Differential Forms, de Rham cohomology and singular cubical homology 56 2.1 Gradient Maps Revisited . 56 2.2 Differential Forms . 60 2.3 De Rham’s Cohomology . 65 2.4 Integration of differential forms and smooth singular homology . 68 2.4.A Singular homology . 71 2.4.B Integration over chains . 74 2.5 Orientation of chains . 80 2.5.A The Stokes Theorem revisited . 82 2.5.B Index of a cycle, topological degree of a continuous map . 87 3 General Constructions of (Co)homology of complexes 91 3.1 Homology and cohomology of complexes . 91 3.2 General homology and cohomology theories . 101 3.2.A Singular homology revisited . 102 3.2.B Reduced singular homology theory . 112 3.2.C Eilenberg-MacLane theorem . 113 3.2.D Various results and consequences of axioms . 115 3.2.E Reduced homology . 121 3.2.F Mayer-Vietoris sequence . 122 3.2.G Compact supports and continuity . 124 3.2.H Eilenberg-Steenrod Theorem . 127 3.3 Differentiable singular chains and the De Rham theorem . 128 ii Spis treści 4 Various construction of homology and cohomology theories of Čech type 134 4.1 Čech-type homology . 134 4.2 Alexander-Spanier cohomology and Massey homology . 139 4.2.A Alexander-Spanier cohomology . 139 4.2.B Massey homology . 144 4.2.C Eilenberg-Steenrod axioms . 144 4.2.D Excision axiom . 148 4.2.E Continuity of the Massey homology . 152 4.3 The Massey homology with compact supports . 152 4.4 The Alexander-Spanier cohomology theory with compact supports . 152 4.5 The de Rham-Čech Theorem . 152 4.6 Final remarks . 152 5 Fiber Bundles and Coverings 153 6 Brouwer Theorem 155 6.1 Lemat Spernera . 155 Introduction We shall use different techniques in these notes. We assume the basic knowledge of algebra, general topology, functional analysis, differential and integral calculus in Banach spaces. A good source of necessary material on these subjects are [1], [2], [3] and [4, 5]. 0.1 General notation If X is a metric space, then its metric is denoted by d (or by dX . If A ⊂ X and " > 0, then B(A; ") := fx 2 X j dA(x) = d(x; A) := infa2A d(a; x) < "g is the open ball around A of radius ", D(A; ") = fx 2 X j dA(x) 6 "g is the closed ball and S(A; r) = fx 2 X j dA(x) = "g. In n n n−1 n particular D (resp. B ) and S ) is the unit closed (resp. closed) and the unit sphere in R , respectively. ∗ Given a Banach space with a norm k · k, then E is the (Banach) s[ace of continuous linear forms (functionals) over E (in case E is a vector space without topology), then this symbol ∗ denotes the space of all linear forms on E, too). Given p 2 E and x 2 E we write hp; xi to 1 ∗ denote the value p(x). If (pn)n=1 is a sequence from E (sometimes we write, not so formally, ∗ ∗ (pn) ⊂ E ) and p 2 E , then pn * p means that (pn) converges weakly to p, i.e., for all x 2 E, limn!1hpn; xi = hp; xi. 0.2 General topology Throughout the text by a space we mean a Hausdorff topological space; maps between spaces are always assumed to be continuous, unless otherwise stated (sometimes we explicitly speak of continuity in order to make necessary distinctions). If X is a space and A ⊂ X, then by cl A (or A), int A and @A := cl n int A we denote the closure, the interior and the (topological) boundary of A, respectively. Paracompactness and partitions of unity: Let U, V be families of subsets of a set X. We say that U refines V (or that U is a refinement of V) and write V ≺ U, if, for any U 2 U, there is V 2 V such that U ⊂ V . A space X is paracompact if any open covering V of X has an open locally finite refinement U, i.e., V ≺ U and any x 2 X has a neighborhood W such that #fU 2 U j U \ W 6= ;g < 1. For instance any compact space is paracompact and any metrizable space is paracompact (the Stone theorem). Let X be a space. A family fλs : X ! Igs2S is a partition of unity if: 2 Introduction 1 (i) the family fsupp λsgs2S is a locally finite (closed) covering of X ( ); P 2 (ii) for all x 2 X, s2S λs(x) = 1 ( ). Let U be a covering of X. We say that a partition of unity fλsgs2S is inscribed into U (or that it refines U) if fsupp λsgs2S refines U, i.e., for any s 2 S, there is Us 2 U such that supp λs ⊂ Us. We say that fλsgs2S is subordinate to U = fUsgs2S if, for any s 2 S, supp λs ⊂ Us. The following remarkable result is well-known. 0.2.1 Existence of partitions of unity: A space X is paracompact if and only if any open covering U of X admits a partition of unity inscribed into it. To prove it, we need the following lemma. 0.2.2 Lemma: Let fUsgs2S be an open locally finite covering of X. There is a an open cover fVsgs2S such that cl Vs ⊂ Us for any s 2 S. Dowód: to be done. The next result, although sometimes very convenient, is less known. 0.2.3 Theorem: Any open covering of a paracompact space admits a subordinate partition of unity. Proof: Let U = fUsgs2S. There is a partition of unity fηaga2A inscribed to U. Hence, for any a 2 A, there is s = r(a) 2 S such that supp ηa ⊂ Us. Therefore we have defined a function −1 r : A ! S. Obviously fr (s)gs2S is a disjoint cover of A. For s 2 S, let X λs(x) = ηa(x); x 2 X; a2r−1(s) −1 if r (s) 6= ; and λs ≡ 0, otherwise. Clearly λs is well-defined since fsupp ηaga2A is a locally finite (closed) cover of X. Therefore λs : X ! [0; 1] is continuous. Let x 2 X and consider a neighborhood U of x such that the set Ax := fa 2 A j U \ supp ηa 6= ;g is finite. We claim that Sx := fs 2 S j U \ supp λs 6= ;g is also finite. Take s 2 Sx and observe that the family −1 −1 fηa ((0; 1])ga2r−1(s) is locally finite and if λs(y) > 0, then there is a 2 r (s) such that ηa(y) > 0. Hence −1 [ −1 λs ((0; 1]) ⊂ ηa ((0; 1]): a2r−1(s) Therefore −1 [ −1 [ −1 [ supp λs := cl λs ((0; 1]) ⊂ cl ηa ((0; 1]) = cl ηa ((0; 1]) = supp ηa: a2r−1(s) a2r−1(s) a2r−1(s) Suppose that Sx is infinite; then it contains a countable set fs1; s2; :::g ⊂ Sx. For each i 2 N, −1 there is ai 2 r (si) such that U \ supp ηai 6= ;. Of course ai 6= aj if i 6= j, i; j 2 N. Moreover fa1; a2; :::g ⊂ Ax: contradiction. Hence fsupp λsgs2S is locally finite. Moreover, for each x 2 X, X X X X λs(x) = ηa(x) = ηa(x) = 1: s2S s2S a2r−1(s) a2A 1 Recall that the support of f : X ! R is the set supp f := cl fx 2 X j f(x) 6= 0g. 2Some authors consider a more general notion of a partition of unity – see e.g. [14]. Wstęp 3 0.2.4 Remark: If X is a metric space, then any open covering U admits an inscribed partition of unity fλsgs2S such that λs is a locally Lipschitz function for all s 2 S. Indeed, let fWsgs2S be a locally finite open covering refining w U. For s 2 S and x 2 X, let 0 gdy x 62 Ws; as(x) = d(x; bd Ws) gdy x 2 Ws: It is easy to see that as : X ! R, s 2 S, is Lipschitz with constant Ls. Let us define −1 λs(x) := as(x)[g(x)] ; x 2 X P where g(x) := s2S as(x). Since fWsg is locally finite, g is a correctly defined nonvanishing 0 continuous function being locally Lipschitz with constant Ls. Therefore λs, s 2 S, is correctly defined and continuous. We claim that λs, 2 S, is locally Lipschitz. Fix s 2 S, x 2 X and s0 2 S such that x 2 Ws0 .
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