Simplicial Structures in Topology

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VI.2 Fundamental Group and Homology 211 Proof. We define the homomorphism ¯φ(g +[G,G]) := φ(g)+[H,H] for every g +[G,G] ∈ G/[G,G]. � Let us go back to the surfaces. The abelianized group of G = π(nT 2 ,∗) is the group generated by the elements together with the set of relations hence, a1,b1,a2,b2,...,an,bn R = {a 1 1b 1 1a −1 1 b−1 1 ...a1nb 1 na −1 n b −1 n ;xyx −1 y −1 |x,y ∈ G}; π(nT 2 ,∗)/[π(nT 2 ,∗),π(nT 2 ,∗)] ∼ = Z 2n . On the other hand, the abelianized group of H = π(nRP 2 ,∗) is the group presented as Gp(S;R) where S = {a1,a2,...,an} R = {a 1 1a 1 1 ...a 1 na 1 n}∪{xyx −1 y −1 |x,y ∈ S}; therefore, the abelianized group of H is an Abelian group generated by the elements together with the relation a1,a2,...,an 2(a1 + a2 + ...+ an)=1 and thus, by setting h = a1 + ...+ an,wehave π(nRP 2 ,∗)/[π(nRP 2 ,∗),π(nRP 2 ,∗)] ∼ = Gp(a1,...,an−1,h;2h) ∼ = Z n−1 × Z2. Since Z2n and Zn−1 ×Z2 are not isomorphic, we conclude that the two fundamental groups π(nT 2 ,∗) and π(nRP2 ,∗) are not isomorphic. By the way, this conclusion proves once more that nT 2 and nRP2 are not homeomorphic. What is interesting to note is that the abelianized groups of the fundamental groups of nT 2 and nRP2 coincide with the corresponding homology groups H1(nT 2 ;Z) and H1(nRP2 ;Z). Hence, it is reasonable to ask whether this is merely a coincidence or it is a fact that holds true for specific types of polyhedra; the answer to this question is found in the following result: (VI.2.2) Theorem. The abelianized group of the fundamental group of a connected polyhedron |K| is isomorphic to H1(|K|;Z).
212 VI Homotopy Groups Proof. Since K is connected, we may neglect writing the base point and simply refer to π(|K|). We define the function ϕ : π(|K|) → H1(|K|;Z) as follows: let [z] be a generator of H1(S 1 ;Z); then, (∀[ f ] ∈ π(|K|))ϕ([ f ]) := H1( f ;Z)([z]). This definition is independent from the representative chosen for the homotopy class [ f ]. We now wish to make sure that ϕ is a group homomorphism. In fact, for every [ f ],[g] ∈ π(|K|), because ϕ([ f ] ν × [g]) = ϕ([σ( f ∨ g)ν]) = ϕ([ f ]) + ϕ([g]) H1(ν;Z) : H1(S 1 ,Z) → H1(S 1 ∨ S 1 ;Z) ∼ = H1(S 1 ;Z) ⊕ H1(S 1 ;Z), [z] ↦→ ([z],[z]); H1( f ∨ g;Z) : H1(S 1 ;Z) ⊕ H1(S 1 ;Z) → H1(|K|;Z) ⊕ H1(|K|;Z), H1( f ∨ g;Z)([z],[z]) = (H1( f ;Z)([z]),H1(g;Z)([z])); H1(σ;Z) : H1(|K|∨|K|;Z) → H1(|K|;Z), (H1( f ;Z)([z]),H1(g;Z)([z])) ↦→ H1( f ;Z)([z]) + H1(g;Z)([z]). The homomorphism ϕ is surjective: in fact, let c = ∑i miσ i 1 be a 1-cycle of the one-dimensional C1(|K|;Z). Wetakeavertex∗of |K| as the base point of π(|K|); for each 1-simplex σ i 1 of c, letσi 1 (0) be its first vertex and σ i 1 (1) its second one; since K is connected, for each 1-simplex σ i 1 of c, there is a path of 1-simplexes λ i 0 (respectively, λ i 1 ) that joins ∗ to σ i 1 (0) (respectively, σ i 1 (1)). The homotopy class of the loop λ i 0 .σ i 1 .(λ i 1 )−1 , obtained by composition, is an element [ fi] ∈ π(|K|,∗); hence, with a suitable triangulation of S 1 ,wemaysaythat ϕ( ν ×i ( fi) mi )=∑ i mi{λ i 0 .σ i 1 .(λ i 1 )−1 }. However, since |K| is connected, each λ i 0 .σ i 1 .(λ i 1 )−1 is homologous to σ i 1 and so, ϕ is surjective. The surjection ϕ : π(|K|) → H1(|K|;Z) is easily extended to a surjection ϕ : π(|K|)/[π(|K|),π(|K|)] → H1(|K|;Z). We now prove that ϕ is injective. Let us suppose [ f ] ∈ π(|K|) to be such that ϕ([ f ]) = 0 ∈ H1(|K|;Z); then, ϕ([ f ]) is a boundary ϕ([ f ]) = ∂2 � ∑ i miσ 2 i � .
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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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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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Page 181 and 182: 164 IV Cohomology Since g and π r
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Page 188 and 189: Chapter V Triangulable Manifolds V.
Page 190 and 191: V.1 Topological Manifolds 173 is a
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Page 194 and 195: V.2 Closed Surfaces 177 we define C
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Page 233 and 234: 216 VI Homotopy Groups The group π
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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
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Simplicial Structures in Topology

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