Simplicial Structures in Topology

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Chapter I Fundamental Concepts I.1 Topology I.1.1 Topological Spaces Let X beagivenset. Atopology on X is a set U of subsets of X satisfying the following properties: A1 /0,X ∈ U; A2 if {Uα | α ∈ J} is a set of elements of U,then � Uα ∈ U; α∈J A3 if {Uα | α = 1,...,n} is a finite set of elements of U,then n� Uα ∈ U. α=1 A topological space or, simply, a space is a set X with a topology. The elements of U are the open sets of X; axioms A1, A2, and A3 above state that /0, X are open sets, that the union of any number of open sets is open, and that the intersection of any finite number of open sets is open. The complement of an open set is a closed set and, therefore, /0 andX are both open and closed sets. A topology U may also be studied (and characterized) through the set of all closed subsets of the topological space. Let C be such a set; by definition, a set C is closed in X if and only if its complement X �C is open in X,andwe write C ∈ C ⇐⇒ X �C ∈ U. Moreover, the set C of all closed subsets satisfies the following properties which characterize any set of closed sets of X: D.L. Ferrario and R.A. Piccinini, Simplicial Structures in Topology, 1 CMS Books in Mathematics, DOI 10.1007/978-1-4419-7236-1 I, © Springer Science+Business Media, LLC 2011
2 I Fundamental Concepts C1 /0,X ∈ C; C2 if {Uα | α = 1,...,n} is any finite set of elements of C,then n� Uα ∈ C α=1 (in other words, any finite union of closed sets is a closed set); C3 if {Uα | α ∈ J} is any set of elements of C,then � Uα ∈ C α∈J (in other words, any intersection of closed sets is a closed set). For any space X,theclosure Y of a subset Y ⊂ X is the intersection of all closed subsets of X which contain Y; it is, obviously, the smallest closed subset containing Y, andY∈C. Theinterior ˚Y of Y is the union of all open subsets of X contained in Y ; it is the largest open set contained in Y ,and˚Y ∈ U. The following lemma is a useful characterization of the closure of a set. (I.1.1) Lemma. Let Y ⊂ X; theny∈ Y if and only if every open set U containing y intersects Y . Proof. In fact, if U ∩Y = /0, then X �U is a closed subset of X which contains Y ; since y belongs to the closure of Y, it follows that y ∈ X �U, a contradiction. Conversely, suppose that there exists a closed set C ⊂ X containing Y and which does not contain y. ThenX �C is an open set, y ∈ X �C, and(X �C) ∩Y = /0, a contradiction. � For any set X, there are two topologies that come immediately to mind: 1. The discrete topology whose open sets are all the subsets of X, thatistosay, Ud = P(X)=2X . 2. The trivial topology with Ub = {/0,X}. Clearly, Ud ⊃ Ub and this fact brings to mind the idea of comparing two topologies on the same set: we shall say that topology U ′ is finer than topology U if U ′ ⊃ U; hence, the discrete topology of X is finer than the trivial one. AsetBof subsets of X is a basis for a topology on X or a basis of open sets for X if: B1 For each x ∈ X,thereisaB∈Bsuch that x ∈ B (that is to say, X = � B∈B B). B2 If B1,B2 ∈ B and B1 ∩B2 �= /0, then for each x ∈ B1 ∩B2 there exists B3 ∈ B such that x ∈ B3 ⊂ B1 ∩ B2. AbasisBgenerates a topology U on X automatically, by requiring that U ⊂ X be open in the topology U if, and only if, for each x ∈ U, there exists B ∈ U such that x ∈ B ⊂ U; formally, U ∈ U ⇐⇒ (∀x ∈ U)(∃B ∈ B) x ∈ B ⊂ U.
Page 2: Canadian Mathematical Society Soci
Page 5 and 6: Dr. Davide L. Ferrario Università
Page 8: Foreword to the English Edition Exc
Page 11 and 12: x Preface same qualitative properti
Page 13 and 14: xii Preface homology groups, also w
Page 16 and 17: Contents I Fundamental Concepts ...
Page 20 and 21: I.1 Topology 3 We need to show that
Page 22 and 23: I.1 Topology 5 Let X be a topologic
Page 24 and 25: I.1 Topology 7 (x, y) (x, −y) Fig
Page 26 and 27: I.1 Topology 9 that f is a homeomor
Page 28 and 29: I.1 Topology 11 We recall that an i
Page 30 and 31: I.1 Topology 13 Cx = � Xj j∈J i
Page 32 and 33: I.1 Topology 15 A Fig. I.5 Example
Page 34 and 35: I.1 Topology 17 with the topology i
Page 36 and 37: I.1 Topology 19 are closed in Y. Th
Page 38 and 39: I.1 Topology 21 is the diagonal of
Page 40 and 41: I.1 Topology 23 (I.1.38) Lemma. Let
Page 42 and 43: I.1 Topology 25 Here is an example.
Page 44 and 45: I.2 Categories 27 5. Prove that a f
Page 46 and 47: I.2 Categories 29 3. For every pair
Page 48 and 49: I.2 Categories 31 If conditions 2.
Page 50 and 51: I.2 Categories 33 Conversely, given
Page 52 and 53: I.2 Categories 35 q: B ⊔C → B
Page 54 and 55: I.2 Categories 37 (I.2.5) Lemma. Gi
Page 56 and 57: I.3 Group Actions 39 I.3 Group Acti
Page 58: I.3 Group Actions 41 S2 = {(x,y,z)
Page 61 and 62: 44 II Simplicial Complexes (0, 1) (
Page 63 and 64: 46 II Simplicial Complexes II.2 Abs
Page 65 and 66: 48 II Simplicial Complexes II.2.1 T
Page 67 and 68: 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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V.3 Poincaré Duality 191 (V.3.3) T
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V.3 Poincaré Duality 193 therefore
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196 VI Homotopy Groups and define t
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198 VI Homotopy Groups is a homotop
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200 VI Homotopy Groups Proof. First
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202 VI Homotopy Groups (VI.1.10) Ex
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204 VI Homotopy Groups (VI.1.13) Th
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206 VI Homotopy Groups to the simpl
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208 VI Homotopy Groups Proof. For e
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210 VI Homotopy Groups Exercises 1.
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212 VI Homotopy Groups Proof. Since
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214 VI Homotopy Groups is precisely
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216 VI Homotopy Groups The group π
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218 VI Homotopy Groups 4. R ′ α
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220 VI Homotopy Groups and � y0 H
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222 VI Homotopy Groups Let f := F(
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224 VI Homotopy Groups be two homot
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226 VI Homotopy Groups (VI.3.16) Re
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228 VI Homotopy Groups where iA is
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230 VI Homotopy Groups f : I n g: I
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232 VI Homotopy Groups 8. Prove tha
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234 VI Homotopy Groups This derives
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236 VI Homotopy Groups Proof. The f
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References 1. J.W. Alexander - A pr
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Index H-space, 219 n-simple, 225 ab
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Index 243 Euclidean, 43 join, 47 su
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Simplicial Structures in Topology

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