Simplicial Structures in Topology

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III.5 Real Projective Spaces 131 0 �� H3(RP 3 ;Z) ∼ = �� Z �� 0 �� 0 �� H2(RP 3 ;Z) �� Z2 �� ∼ H1(RP = 3 ;Z) which easily leads to the thesis. � At this stage, the reader could be tempted to compute the homology of RP 4 with this method and then apply the Exact Sequence Theorem in Homology to compute the homology of RP n by induction; unfortunately, this idea will not work, because the connecting homomorphisms λq : Hq(S n ,Z) → Hq−1(RP n−1 ,Z) are generally not known. Therefore, we must come up with another method for computing the homology of RP n when n ≥ 4; this will be done in the next subsection. III.5.1 Block Homology We begin by reviewing how to compute the homology of the torus T 2 . To start with, we interpret the torus T 2 as deriving from the square with vertices (0,0), (1,0), (0,1), and(1,1) by identifying the two horizontal edges and then the two vertical ones. A possible triangulation T of the torus is represented in Fig. III.4 0 1 2 0 3 5 6 3 4 7 8 4 0 1 2 0 Fig. III.4 (see also p. 63). We have 9 vertices, 27 edges, and 18 faces, a total of 54 simplexes in it, too large a number, considering what we started with; in fact, given the identifications made on the Euclidean square, we could take the simplexes in “blocks”, consider only one vertex (the four vertices have been identified), two onedimensional “blocks”, namely, one vertical edge and one horizontal edge (without their end-points), and only one two-dimensional “block”, namely, the square without its boundary. We then ask ourselves whether it is possible to compute the homology of T 2 by means of this block (or “cellular”) interpretation of the torus and, in doing so, avoid handling a rather large number of simplexes; because, if the procedure is already a little complicated for the torus, a space that can, after �� 0,
132 III Homology of Polyhedra all, be easily interpreted, imagine what might happen when we try to compute the homology of a more complex polyhedron. We now explain what a “block” decomposition of a polyhedron is and how the homology of a polyhedron divided into blocks is computed. Let K =(X,Φ) be a given simplicial complex; for every subset e ⊂ Φ, letebe the smallest subcomplex of K that contains e (we observe that e is not necessarily a simplicial complex; for instance, take the previous triangulation T of the torus T 2 and let e be the set of the simplexes {2}, {0,3,5}, and{3,4,6}; in this example, {0,3} ⊂{0,3,5} but {0,3} �∈ e). The simplicial complex e may be described in another way: e = � σ . The simplicial complex e associated with the set e is called closure of e. For every e ⊂ Φ, wedefinetheboundary of e as σ∈e • e = e � e; this, by definition, is a subcomplex of K; furthermore, • e is a simplicial subcomplex of e: e � e ⊂ e ⇒ e � e ⊂ e = e ⇒ • e ⊂ e . Finally, e = e ∪ • e: in fact, since e ⊂ e and • e ⊂ e, we have the inclusion e ∪ • e ⊂ e; on the other hand, if σ ∈ e and σ �∈ e then σ ∈ e � e and thus σ ∈ e � e = • e ; we conclude that e ∪ • e ⊂ e. We extend the definition of closure of a set e ⊂ Φ to a chain cn = ∑i miσ i n ∈ Cn(K,Z): theclosure of cn is the simplicial complex σ i : mi�=0 i n . cn = � (III.5.3) Definition. A p-block or block of dimension p of K is a set ep ⊂ Φ such that: (i) ep contains no simplex of dimension > p (ii) Hp(ep, • ep;Z) ∼ = Z (iii) (∀q �= p) Hq(ep, • ep;Z) ∼ = 0 (III.5.4) Remark. To compute the homology of the pair of simplicial complexes (ep, • ep), it is necessary to give an orientation to K. However, we also could give an orientation to ep individually by choosing a generator βp of the Abelian group Hp(ep, • ep;Z) ∼ = Zp(ep, • ep) ∼ = Z .
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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
Page 97 and 98: 80 II Simplicial Complexes Φi = {
Page 99 and 100: 82 II Simplicial Complexes obtained
Page 101 and 102: 84 II Simplicial Complexes (that is
Page 103 and 104: 86 II Simplicial Complexes Therefor
Page 105 and 106: 88 II Simplicial Complexes Exercise
Page 107 and 108: 90 II Simplicial Complexes Hn(C;G)=
Page 109 and 110: 92 II Simplicial Complexes Since im
Page 111 and 112: 94 II Simplicial Complexes and cons
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Page 116 and 117: Chapter III Homology of Polyhedra I
Page 118 and 119: III.1 The Category of Polyhedra 101
Page 126 and 127: III.2 Homology of Polyhedra 109 Now
Page 128 and 129: III.2 Homology of Polyhedra 111 whe
Page 130 and 131: III.2 Homology of Polyhedra 113 A s
Page 132 and 133: III.2 Homology of Polyhedra 115 Fro
Page 134 and 135: III.3 Some Applications 117 where H
Page 136 and 137: III.3 Some Applications 119 we now
Page 138 and 139: III.3 Some Applications 121 has no
Page 140 and 141: III.3 Some Applications 123 Proof.
Page 142 and 143: III.4 Relative Homology 125 6. Let
Page 144 and 145: III.4 Relative Homology 127 such th
Page 146 and 147: III.5 Real Projective Spaces 129 We
Page 150 and 151: III.5 Real Projective Spaces 133 No
Page 152 and 153: III.5 Real Projective Spaces 135 Le
Page 154 and 155: III.5 Real Projective Spaces 137 he
Page 156 and 157: III.5 Real Projective Spaces 139 We
Page 158 and 159: III.6 Homology of the Product of Tw
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Page 169 and 170: 152 IV Cohomology (IV.1.1) Theorem.
Page 171 and 172: 154 IV Cohomology (IV.1.2) Remark.
Page 173 and 174: 156 IV Cohomology If, starting from
Page 175 and 176: 158 IV Cohomology When we apply thi
Page 177 and 178: 160 IV Cohomology groups Q and R ar
Page 179 and 180: 162 IV Cohomology (IV.2.2) Corollar
Page 181 and 182: 164 IV Cohomology Since g and π r
Page 183 and 184: 166 IV Cohomology It is easily prov
Page 185 and 186: 168 IV Cohomology ∩: B p (K;Z) ×
Page 188 and 189: Chapter V Triangulable Manifolds V.
Page 190 and 191: V.1 Topological Manifolds 173 is a
Page 192 and 193: V.1 Topological Manifolds 175 |Ki|
Page 194 and 195: V.2 Closed Surfaces 177 we define C
Page 196 and 197: 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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