Simplicial Structures in Topology

More documents

Recommendations

Info

II.4 Simplicial Homology 83 � x − v if x �= v, ∂1s0(x)= 0 ifx = v. Case 2: n > 0–Letσ be any oriented n-simplex of vK. We first observe that ιCn(c)(σ)=0; moreover, if v is not a vertex of σ,wehave Consequently, v �∈ σ implies ∂n+1(vσ)=σ − v∂n(σ). sn−1∂n(σ)+∂n+1sn(σ)=v∂n(σ)+∂n+1(vσ)=σ. We now suppose that v ∈ σ. Then, sn−1∂n(σ)+∂n+1sn(σ)=sn−1∂n(σ)=σ. It follows from these remarks that ιC(c) and the identity homomorphism 1 C(vK) are homotopic and we conclude from Proposition (II.3.4) that, for every n ∈ Z, Hn(ι)Hn(c) coincides with the identity homomorphism. � We now seek a better understanding of the relative homology H∗(K,L;Z) of a pair of simplicial complexes (K,L). As usual, K =(X,Φ) and L =(Y,Ψ) with Y ⊂ X and Ψ ⊂ Φ. Let v be a point which is not in the set of vertices of either K or L. Let CL be the abstract cone vL. It follows from the definitions that K ∩CL = L. (II.4.7) Theorem. The homology groups Hn(K,L;Z) and Hn(K ∪CL;Z) are isomorphic for each n ≥ 1. Proof. The central idea in this proof is to compare the exact homology sequence of the pair (K,L) and the exact sequence of Mayer–Vietoris of K and CL, before using the Five Lemma; the notation is the one already adopted for the Mayer–Vietoris Theorem. Let us consider the simplicial function f : K → CL defined on the vertices by � x if x ∈ Y f (x)= v if x ∈ X �Y. For each nonnegative integer n, we now define the homomorphisms and ˜kn : Cn(K) → Cn(K) ⊕Cn(CL) , c ↦→ (c,Cn( f )(c)) ˜hn : Cn(K)/Cn(L) −→ Cn(K ∪CL), c +Cn(L) ↦→ Cn( j1)(c) −Cn( j2)Cn( f )(c)
84 II Simplicial Complexes (that is to say, ˜hn(c+Cn(L)) = ˜jn ˜kn(c)). The function ˜hn is well defined; in fact, had c ′ ∈ Cn(K) been such that c−c ′ ∈ Cn(L), we would have c−c ′ ∈ Cn(K ∩CL) and so ˜jn ˜kn(c − c ′ )=˜jnĩn(c − c ′ )=0. The homomorphism sequences ˜h = {˜hn|n ≥ 0} and ˜k = {˜kn|n ≥ 0} are homomorphisms of chain complexes giving rise to a commutative diagram C(L) �� C(i) �� 1 ˜k C(K ∩CL) �� ĩ C(K) ⊕C(CL) �� C(K) ˜q ˜j �� C(K,L) ˜h �� C(K ∪CL) Since CL is an acyclic simplicial complex, we obtain, for every n ≥ 2, the commutative diagram of Abelian groups Hn(L;Z) 1 �� Hn(L;Z) �� Hn(K;Z) ∼= �� Hn(K;Z) �� Hn(K,L;Z) γ �� Hn(K ∪CL;Z) �� Hn−1(L;Z) 1 �� Hn−1(L;Z) �� Hn−1(K;Z) ∼= �� Hn−1(K;Z) and by the Five Lemma, we conclude that γ is an isomorphism; when n = 1, the last vertical arrow is an injective homomorphism H0(K;Z) −→ H0(K;Z) ⊕ Z and again with an argument similar to the Five Lemma, we conclude that γ is an isomorphism. � (II.4.8) Remark. We recall that we have defined the relative homology groups of a pair of simplicial complexes (K,L) through the chain complex C(K,L) = {Cn(K)/Cn(L),∂ K,L n }; we now construct the relative groups Hn(K,L;Z), n ≥ 0, from a slightly different point of view which turns out to be very useful for computing homology groups. For any n ≥ 0, let Cn(K,L) be the Abelian group of formal linear combinations, with coefficients in Z, ofalln-simplexes of K which are not in L; in other words, if K =(X,Φ), L =(Y,Ψ) with Y ⊂ X and Ψ ⊂ Φ, Cn(K,L;Z)={∑ i miσ i n |σ i n ∈ Φ �Ψ}. The inclusion i: L → K induces an injective homomorphism Cn(i): Cn(L) → Cn(K) for each n ≥ 0; we now take, for every n ≥ 0, the following linear homomorphisms: βn : Cn(K) → Cn(L)
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 18 and 19:
Chapter I Fundamental Concepts I.1
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
Page 69 and 70: 52 II Simplicial Complexes r = tp+(
Page 71 and 72: 54 II Simplicial Complexes In a sim
Page 73 and 74: 56 II Simplicial Complexes (not nec
Page 75 and 76: 58 II Simplicial Complexes Let K3,3
Page 77 and 78: 60 II Simplicial Complexes ordering
Page 79 and 80: 62 II Simplicial Complexes are two
Page 81 and 82: 64 II Simplicial Complexes 1-simple
Page 83 and 84: 66 II Simplicial Complexes An infin
Page 85 and 86: 68 II Simplicial Complexes 2. λn i
Page 87 and 88: 70 II Simplicial Complexes Please n
Page 89 and 90: 72 II Simplicial Complexes (II.3.7)
Page 91 and 92: 74 II Simplicial Complexes (II.3.10
Page 93 and 94: 76 II Simplicial Complexes If we do
Page 95 and 96: 78 II Simplicial Complexes In the c
Page 97 and 98: 80 II Simplicial Complexes Φi = {
Page 99: 82 II Simplicial Complexes obtained
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
Page 113 and 114: 96 II Simplicial Complexes In parti
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 148 and 149: III.5 Real Projective Spaces 131 0
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
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166:
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
Page 202 and 203:
V.2 Closed Surfaces 185 Before proc
Page 204 and 205:
V.2 Closed Surfaces 187 Simplicial
Page 206 and 207:
V.3 Poincaré Duality 189 to the co
Page 208 and 209:
V.3 Poincaré Duality 191 (V.3.3) T
Page 210:
V.3 Poincaré Duality 193 therefore
Page 213 and 214:
196 VI Homotopy Groups and define t
Page 215 and 216:
198 VI Homotopy Groups is a homotop
Page 217 and 218:
200 VI Homotopy Groups Proof. First
Page 219 and 220:
202 VI Homotopy Groups (VI.1.10) Ex
Page 221 and 222:
204 VI Homotopy Groups (VI.1.13) Th
Page 223 and 224:
206 VI Homotopy Groups to the simpl
Page 225 and 226:
208 VI Homotopy Groups Proof. For e
Page 227 and 228:
210 VI Homotopy Groups Exercises 1.
Page 229 and 230:
212 VI Homotopy Groups Proof. Since
Page 231 and 232:
214 VI Homotopy Groups is precisely
Page 233 and 234:
216 VI Homotopy Groups The group π
Page 235 and 236:
218 VI Homotopy Groups 4. R ′ α
Page 237 and 238:
220 VI Homotopy Groups and � y0 H
Page 239 and 240:
222 VI Homotopy Groups Let f := F(
Page 241 and 242:
224 VI Homotopy Groups be two homot
Page 243 and 244:
226 VI Homotopy Groups (VI.3.16) Re
Page 245 and 246:
228 VI Homotopy Groups where iA is
Page 247 and 248:
230 VI Homotopy Groups f : I n g: I
Page 249 and 250:
232 VI Homotopy Groups 8. Prove tha
Page 251 and 252:
234 VI Homotopy Groups This derives
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
show all

Simplicial Structures in Topology

Create successful ePaper yourself

Delete template?

Save as template?