Simplicial Structures in Topology

More documents

Recommendations

Info

II.4 Simplicial Homology 85 defined on the n-simplexes of K by the conditions such that � 0 if σn ∈ Φ �Ψ βn(σn)= σn if σn ∈ Ψ αn : Cn(K,L) → Cn(K) , σn ∈ Φ �Ψ ↦→ σn μn : Cn(K) → Cn(K,L) � σn if σn ∈ Φ �Ψ μn(σn)= 0 if σn ∈ Ψ . It is easy to check that βnCn(i)=1, μnαn = 1, μnCn(i)=0, and Cn(i)βn + αnμn = 1 for each n ≥ 0. Hence, for every n ≥ 0, we have a short exact sequence Cn(L) �� Cn(i) �� Cn(K) μn �� Cn(K,L). We now consider the boundary homomorphism ∂n : Cn(K) → Cn−1(K) and define ∂ n : Cn(K,L) → Cn−1(K,L) as the composite homomorphism ∂ n = μn−1∂nαn. We note that ∂ n−1∂ n =(μn−2∂n−1αn−1)(μn−1∂nαn) =(μn−2∂n−1)(1 −Cn−1(i)βn−1)∂nαn = μn−2∂n−1∂nαn − μn−2∂n−1Cn−1βn−1 = 0 since the factor ∂n−1∂n = 0 appears in the first term and also because the second term is null on all (n − 1)-simplex of K. The graded Abelian group {Cn(K,L) | n ∈ Z}, where Cn(K,L)=0foreveryn < 0, has a boundary homomorphism {∂ n | n ∈ Z} with ∂ n = 0forn ≤ 0; let H∗(K,L;Z)={Hn(K,L;Z)} be its homology. Let θn : Cn(K,L) → Cn(K) be the linear homomorphism defined on an n-simplex σn ∈ Φ �Ψ by θn(σn)=σn +Cn(L) (if n < 0, we define θn = 0). We note that θn commutes with the boundary homomorphisms; it is sufficient to verify this statement for an n-simplex σn ∈ Φ �Ψ: ∂ K,L n θn(σn)=∂n(σn)+Cn−1(L)= ∑ (−1) σn−1,i∈Φ�Φ i σn−1,i +Cn(L); θn−1∂ n(σn)=θn−1(μn−1∑ i (−1) i σn−1,i)= ∑ (−1) σi,n−1∈Φ�Φ i σn−1,i +Cn(L).
86 II Simplicial Complexes Therefore, the set {θn | n ∈ Z} induces a homomorphism Hn(θn): Hn(K,L;Z) → Hn(K,L;Z) . On the other hand, θn is an isomorphism for each n ≥ 0 (it is injective by definition and surjective because θnμn = qn). Therefore, the two types of homology groups are isomorphic. Let {Ki =(Xi,Φi) |i = 1,...,p} be a finite set of simplicial complexes; we choose a base vertex x i 0 ∈ Xi for each Ki and construct the wedge sum of all Ki as the simplicial complex that is to say ∨ p i=1Ki n� : = i=1 ({x 1 0 }×...× Ki × ...×{x p ∨ p i=1Ki =(∨ p p i=1Xi,∨ i=1Φi). The next theorem shows that the homology of the wedge sum of simplicial complexes acts in a special way. (II.4.9) Theorem. For every q ≥ 1, Hq(∨ p i=1 Ki;Z) ∼ = ⊕ p i=1 Hq(Ki;Z). Proof. It is enough to prove this result for p = 2. The short exact sequence of chain complexes C(K1) �� i �� C(K1 ∨ K2) induces a long exact sequence of homology groups 0 }) k �� C(K1 ∨ K2,K1;Z) ...→ Hn(K1;Z) Hn(i) −→ Hn(K1 ∨ K2;Z) q∗(n) −→ Hn(K2;Z) λn −→ Hn−1(K1;Z) → ... �� (see Remark (II.4.8)). complexes Let us now examine how the homomorphisms of chain C(K1) i C(K1 ∨ K2) C(K1 ∨ K2) k ��C(K1 ∨ K2,K1;Z) ∼ = C(K2) are defined on simplexes (in other words, the generators of the free groups that concern us): (∀σ 1 n ∈ Φ1) in(σn)=σ 1 n ×{x2 0 } (∀σ 1 n ×{x 2 0}∈Φ1 ×{x 2 0}) kn(σ 1 n ×{x 2 0} = 0 (∀{x 1 0 }×σ 2 n ∈{x1 0 }×Φ2) kn({x 1 0 }×σ 2 n )=σ 2 n .
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 and 100: 82 II Simplicial Complexes obtained
Page 101: 84 II Simplicial Complexes (that is
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?