Simplicial Structures in Topology

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Index H-space, 219 n-simple, 225 abstract cone, 47 action of a group, 39 antipodal point, 15 barycentric subdivision, 102 Betti numbers, 75 block (of dimension p), 132 block homology, 131 block triangulation, 133 boundary, 61 boundary homomorphism, 60 Brouwer Fixed Point Theorem, 121 cap product, 166 category, 27 CTop, 28 Set, 28 Set∗, 28 Ab, 28 Ab Z , 28 Gr, 28 Top, 28 Top ∗, 28 morphism, 27 object, 27 product, 28 simplicial complexes, 47 chain, 60 cycle, 61 homologous, 61 homotopy, 69 null-homotopic, 69 chain carrier, 73 chain complex, 65 acyclic, 70 augmented, 70 free, 70 positive, 70 closed surfaces, 176 Fundamental Theorem, 184 closure, 2 cochain, 155 difference, 235 cocycle, 155 obstruction, 234 cohomology, 153 of polyhedra, 158 ring, 162 commutator, 210 complex cochain, 155 comultiplication, 196 associative, 196 connected simply, 224 connected sum, 178 covering, 16 refinement, 16 cup product, 160 distance, 3, 25 Euler–Poincaré Characteristic, 76 exact sequence, 66 homology, 77 short, 66 split, 75 fibration, 231 Five Lemma, 74 folding map, 196 241
242 Index free presentation, 93 function adjoint, 22 continuous, 7 evaluation, 23 simplicial, 47 functor adjoint, 33 contravariant, 31 covariant, 30 equivalent, 33 forgetful, 31 geometric realization, 49 suspension, 31 universal elements, 141 with models, 141 fundamental group, 198 Fundamental Theorem of Algebra, 122 geometric realization, 48 group cohomology, 153 divisible, 159 homotopy, 216 locally free, 95 homeomorphism, 9 homology product of two polyhedra, 147 reduced, 87 relative, 76 homology groups, 61 homomorphism augmentation, 70 adjoint, 151 chain, 65 coboundary, 155 connecting, 67, 77 natural, 69 homotopy, 29 chain, 69 equivalence, 30 relative, 30 type, 30 Homotopy Extension Property, 105 homotopy invariant, 115 Hopf Trace Theorem, 119 interior, 2 interval closed, 11 open, 11 semi-open, 11 Lebesgue number, 24 Lefschetz Fixed Point Theorem, 119 Lefschetz number, 118 loop, 195 manifold n-dimensional, 171 orientable, 189 triangulable, 173 map, 7 closed, 18 degree, 226 open, 27 quotient, 9 metric, 24 morphism codomain, 27 domain, 27 multiplication, 196 neighbourhood, 7 open disk, 3 orbit, 40 polyhedron, 55 diameter, 104 product extension, 157 tensor, 89 torsion, 93 pullback, 230 pushout, 33 real projective plane, 5 retract deformation, 38 strong, 38 set closed, 1 open, 1 simplex abstract, 46 boundary • σ, 47 closure, 47 Euclidean, 43 face, 46 geometric, 51 simplicial approximation, 113 simplicial complex abstract, 46 connected, 80 dimension, 46
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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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V.2 Closed Surfaces 177 we define C
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V.2 Closed Surfaces 179 Fig. V.6 Co
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V.2 Closed Surfaces 181 a a b a a d
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V.2 Closed Surfaces 183 where i = 1
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V.2 Closed Surfaces 185 Before proc
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V.2 Closed Surfaces 187 Simplicial
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V.3 Poincaré Duality 189 to the co
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Page 217 and 218: 200 VI Homotopy Groups Proof. First
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Page 229 and 230: 212 VI Homotopy Groups Proof. Since
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Page 233 and 234: 216 VI Homotopy Groups The group π
Page 235 and 236: 218 VI Homotopy Groups 4. R ′ α
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Page 249 and 250: 232 VI Homotopy Groups 8. Prove tha
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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 260: Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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