Simplicial Structures in Topology

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VI.3 Homotopy Groups 227 Since the map 1S n : Sn → S n has obviously degree 1, the degree function that we have defined is surjective. It must be proved that the degree function is injective; we do not prove it here, but the reader can find the proof in the work of H. Whitney [34]. However, Theorem (VI.3.17) may also be proved in different ways (see, for instance, Sect. V.3 of [26]). � Lemma (VI.3.18) proves Theorem (VI.3.17). So far, not all homotopy groups of the spheres are known; yet, we have two important results due to Jean–Pierre Serre [31] and based on difficult techniques of homological algebra: (VI.3.19) Theorem. If n is odd and m �= n, then πm(Sn ,e0) is a finite group. (VI.3.20) Theorem. If n is even, then 1. πm(Sn ,e0) is finite if m �= n and m �= 2n − 1. 2. π2n−1(Sn ,e0) is the direct sum of an infinite cyclic group and a finite group (eventually trivial). VI.3.3 Another Approach to Homotopy Groups In the literature, the homotopy groups are also described in a different way but equivalent to ours; sometimes it is convenient to study the homotopy groups under this other point of view. We begin our work with a lemma that is very important to the development of our theme. (VI.3.21) Lemma. Given two maps f ,g ∈ CTop((X,A),(Y,B)) such that f |A = g|A, let H : (X × I,A × I) −→ (Y,B) be a homotopy relative to A from f to g. Then, there exists a based homotopy between the maps induced by f and g. H : X/A × I −→ Y /B f ,g: X/A −→ Y /B Proof. The quotient space X/A is given by the following pushout cA A �� ∗ iA �� X qX �� X/A
228 VI Homotopy Groups where iA is the inclusion map and qX : X → X/A is a quotient map. By Corollary (I.1.40), also the following diagram is a pushout: A × I �� ∗×I iA × 1I �� X × I qX × 1I �� X/A × I Let qY : Y →Y/B be the quotient map and c: ∗×I →Y/B be the constant map at the base point of Y/B. Since(qY H)(iA ×1I)=c(cA ×1I), by the Universal Property of Pushouts, there exists a homotopy H : X/A × I −→ Y /B with the required properties. � We now choose a special pair: for every n ≥ 1, (I n ,∂I n ) is the pair defined by the n-dimensional hypercube and its boundary ∂I n . Given any map f ∈ CTop((I n ,∂I n ),(Y,y0)), we consider the following pushout diagram ∂ I n ι �� q �� ∗ ι �� q I n �� f �� f �� Y where ι is the inclusion map and q is a quotient map qIn followed by the homeomorphism In /∂I n ∼ = Sn . By the preceding lemma, if S n g ∈ CTop((I n ,∂I n ),(Y,y0)) is homotopic to f rel∂I n ,thenf and g are homotopic through a based homotopy. Let [(I n ,∂I n ),(Y,y0)] rel∂ I n := CTop((I n ,∂ I n ),(Y,y0))/rel∂I n be the set of the homotopy classes rel∂I n of maps of pairs from (I n ,∂I n ) to (Y,y0). The following result is easily obtained from our preceding remarks.
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Canadian Mathematical Society Soci
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Dr. Davide L. Ferrario Università
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Foreword to the English Edition Exc
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x Preface same qualitative properti
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xii Preface homology groups, also w
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Contents I Fundamental Concepts ...
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Chapter I Fundamental Concepts I.1
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I.1 Topology 3 We need to show that
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I.1 Topology 5 Let X be a topologic
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I.1 Topology 7 (x, y) (x, −y) Fig
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I.1 Topology 9 that f is a homeomor
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I.1 Topology 11 We recall that an i
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I.1 Topology 13 Cx = � Xj j∈J i
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I.1 Topology 15 A Fig. I.5 Example
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I.1 Topology 17 with the topology i
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I.1 Topology 19 are closed in Y. Th
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I.1 Topology 21 is the diagonal of
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I.1 Topology 23 (I.1.38) Lemma. Let
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I.1 Topology 25 Here is an example.
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I.2 Categories 27 5. Prove that a f
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I.2 Categories 29 3. For every pair
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I.2 Categories 31 If conditions 2.
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I.2 Categories 33 Conversely, given
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I.2 Categories 35 q: B ⊔C → B
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I.2 Categories 37 (I.2.5) Lemma. Gi
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I.3 Group Actions 39 I.3 Group Acti
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I.3 Group Actions 41 S2 = {(x,y,z)
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44 II Simplicial Complexes (0, 1) (
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46 II Simplicial Complexes II.2 Abs
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48 II Simplicial Complexes II.2.1 T
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50 II Simplicial Complexes Let us r
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52 II Simplicial Complexes r = tp+(
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54 II Simplicial Complexes In a sim
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56 II Simplicial Complexes (not nec
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58 II Simplicial Complexes Let K3,3
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60 II Simplicial Complexes ordering
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62 II Simplicial Complexes are two
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64 II Simplicial Complexes 1-simple
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66 II Simplicial Complexes An infin
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68 II Simplicial Complexes 2. λn i
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70 II Simplicial Complexes Please n
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72 II Simplicial Complexes (II.3.7)
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74 II Simplicial Complexes (II.3.10
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76 II Simplicial Complexes If we do
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78 II Simplicial Complexes In the c
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80 II Simplicial Complexes Φi = {
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82 II Simplicial Complexes obtained
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84 II Simplicial Complexes (that is
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86 II Simplicial Complexes Therefor
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88 II Simplicial Complexes Exercise
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90 II Simplicial Complexes Hn(C;G)=
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92 II Simplicial Complexes Since im
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94 II Simplicial Complexes and cons
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96 II Simplicial Complexes In parti
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Chapter III Homology of Polyhedra I
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III.1 The Category of Polyhedra 101
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III.2 Homology of Polyhedra 109 Now
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III.2 Homology of Polyhedra 111 whe
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III.2 Homology of Polyhedra 113 A s
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III.2 Homology of Polyhedra 115 Fro
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III.3 Some Applications 117 where H
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III.3 Some Applications 119 we now
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III.3 Some Applications 121 has no
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III.3 Some Applications 123 Proof.
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III.4 Relative Homology 125 6. Let
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III.4 Relative Homology 127 such th
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III.5 Real Projective Spaces 129 We
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III.5 Real Projective Spaces 131 0
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III.5 Real Projective Spaces 133 No
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III.5 Real Projective Spaces 135 Le
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III.5 Real Projective Spaces 137 he
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III.5 Real Projective Spaces 139 We
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III.6 Homology of the Product of Tw
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152 IV Cohomology (IV.1.1) Theorem.
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154 IV Cohomology (IV.1.2) Remark.
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156 IV Cohomology If, starting from
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158 IV Cohomology When we apply thi
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160 IV Cohomology groups Q and R ar
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162 IV Cohomology (IV.2.2) Corollar
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164 IV Cohomology Since g and π r
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166 IV Cohomology It is easily prov
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168 IV Cohomology ∩: B p (K;Z) ×
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Chapter V Triangulable Manifolds V.
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V.1 Topological Manifolds 173 is a
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V.1 Topological Manifolds 175 |Ki|
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Page 204 and 205: V.2 Closed Surfaces 187 Simplicial
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Page 217 and 218: 200 VI Homotopy Groups Proof. First
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Page 233 and 234: 216 VI Homotopy Groups The group π
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Page 253 and 254: 236 VI Homotopy Groups Proof. The f
Page 256 and 257: References 1. J.W. Alexander - A pr
Page 258 and 259: Index H-space, 219 n-simple, 225 ab
Page 260: Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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