Simplicial Structures in Topology

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III.5 Real Projective Spaces 139 We find a triangulation Mn−1 of RPn−1 in a similar way and so on. Note that • en coincides with the set of (n − 1)-simplexes en−1 of Mn−1 , which in turn is an (n − 1)-block, etc. In this way, we obtain a block triangulation of RPn . All that is left to do is to look into the generators of Hr(er, • er;Z) and their boundaries. We denote a generator of Hn−1(Sn−1 ,Z) with zn−1 ∈ Zn−1(Sn−1 ;Z). 6 The exact sequence of the homology groups of (Kn +,K n−1 ) shows that zn−1 ∗ an+1 is a generating cycle of Hn(Kn + ,Kn−1 ;Z). It follows that ℵ(zn−1 ∗ an+1) is a generator for Zn((Kn + )(1) ,(K n−1 ) (1) ;Z) and if q : |Kn + |→|Mn | is the quotient map, then qℵ(zn−1 ∗ an+1) is a generator of Zn(|Mn |,|Mn−1 |;Z). Now, ∂qℵ(zn−1 ∗ an+1)=qℵ∂(zn−1 ∗ an+1)= (−1) n qℵ(zn−1)=(−1) n qℵ(zn−2 ∗ an − zn−2 ∗ a ′ n) . We interchange in zn−2 all points ar with a ′ r; this gives rise to an element which we denote with z ′ n−2 ; indeed, zn−2 =(−1) n−1z ′ n−2 . Hence, ∂qℵ(zn−1 ∗ an+1)=(−1) n qℵ(zn−2 ∗ an − (−1) n−1 z ′ n−2 ∗ a′ n ) =(−1) n qℵ(zn−2 ∗ an) − (−1) n (−1) n−1 qℵ(z ′ n−2 ∗ a ′ n) . Finally, since qℵ(zn−2 ∗ an+1)=qℵ(z ′ n−2 ∗ a′ n), we conclude that ∂qℵ(zn−1 ∗ an+1)=(1 +(−1) n )qℵ(zn−2 ∗ an) . To simplify the notation, we write βr = qℵ(zr−1 ∗ ar+1); this is a generator of Cr(e(M n );Z) and has the property ∂(βr)=(1 +(−1) r )βr−1 . Specifically for 0 < 2r ≤ n, the group C2r(e(M n );Z) is generated by the elements β2r and ∂(β2r) =2β2r−1; therefore, Z2r(e(Mn );Z) =0andH2r(RPn ;Z) =0. If 2r −1 < n, then the group C2r−1(e(Mn );Z) is generated by β2r−1 and ∂(β2r−1)=0; hence, the group Z2r−1(e(Mn );Z) ∼ =Z is generated by β2r−1, the group B2r−1(e(Mn ); Z) is generated by 2β2r−1, and there is an isomorphism H2r−1(RPn ;Z) ∼ = Z2. We have proved the following result: (III.5.10) Theorem. Hp(RP n ,Z) ∼ ⎧ ⎪⎨ Z if p = 0 0 if 0 inez1 as the join of z0 and {a1,a ′ 1 },etc.
140 III Homology of Polyhedra III.6 Homology of the Product of Two Polyhedra In this section, we study the homology of the product of two polyhedra. The main result is given by the Acyclic Models Theorem. III.6.1 Acyclic Models Theorem This theorem is due to Samuel Eilenberg and Saunders MacLane (see [12]). The reader who wishes to go considerably deeply in this subject is advised to consult the book [4] by Michael Barr. In former sections, we have noticed that we can associate a chain complex C(K) with an augmentation ε : C0(K) → Z , Σ n i=1ai{xi} ↦→ Σ n i=1ai to every oriented simplicial complex K. We have also proved the Acyclic Carrier Theorem (II.3.9) which makes it possible to compare two augmented chain complexes under certain conditions (broadly speaking, these conditions require that some of the local homology groups be trivial). Unfortunately, the Acyclic Carrier Theorem is not powerful enough for studying the homology of a product of two polyhedra; to this end, we need the Acyclic Models Theorem. We have defined the category C of chain complexes in Sect. II.3; in this section, we work with a subcategory of C, namely, the category Clp whose objects are chain complexes of free Abelian groups with an augmentation homomorphism; as we have done in C, we indicate the objects of Clp with sequences of free Abelian groups ··· �� Cn ∂n �� Cn−1 ∂n−1 �� ··· ∂1 �� C0 and a morphism between objects C and C ′ with a commutative diagram ··· ··· �� Cn �� ′ C n fn ∂n �� Cn−1 �� ∂ ′ n �� ′ C n−1 fn−1 Let a category C and a (covariant) functor ∂n−1 �� ··· ∂n−1 �� ··· F : C −→ Clp ∂1 �� C0 �� ∂1 �� ′ C 0 f0 ε �� Z ε �� Z ε ′ �� Z ¯f
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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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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
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Page 146 and 147: III.5 Real Projective Spaces 129 We
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Page 150 and 151: III.5 Real Projective Spaces 133 No
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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
Page 198 and 199: V.2 Closed Surfaces 181 a a b a a d
Page 200 and 201: V.2 Closed Surfaces 183 where i = 1
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Page 204 and 205: 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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