Simplicial Structures in Topology

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I.1 Topology 11 We recall that an interval of R (or more generally, of an ordered set) is a subset A ⊂ R such that, for each a,b,x ∈ R with a < x < b, ifa,b ∈ A, wehavex ∈ A. Real line intervals are important examples of connected spaces, as shown by the next theorem. It is not difficult to prove that every (non-empty) interval of R must be of one of the following types. 2 The set [a,b]={x ∈ R | a ≤ x ≤ b} is the closed interval with end-points a,b;theset (a,b)={x ∈ R | a < x < b} is the open interval with end-points a,b;thesets (a,b]={x ∈ R | a < x ≤ b} and [a,b)={x ∈ R | a ≤ x < b} are semi-open intervals. Thesets (a,+∞)={x ∈ R | a < x}, [a,+∞)={x ∈ R | a ≤ x}, (−∞,b)={x ∈ R | x < b} e (−∞,b]={x ∈ R | x ≤ b} are infinite intervals (and naturally R =(−∞,∞) is the maximal interval). (I.1.14) Theorem. Any interval of R is a connected space. Proof. Let X ⊂ R be a closed interval, say X =[a,b]. LetU �= /0 be a subset of X, both open and closed; we wish to prove that U = X. SinceU is open in X, wemay choose u ∈ U such that u ∈ ˚X. Let s = sup{x ∈ X | [u,x) ⊂ U}; clearly, u < s. We prove that [u,s) ⊂ U. Indeed, for each v ∈ [u,s), there exists x ∈ X such that v < x and [u,x) ⊂ U; hence, v ∈ U. We now prove that s = b. In fact, if s �= b, thens < b; sinceU is closed in X, we conclude that s ∈ U and so, [u,s] ⊂ U. However, U is also open and consequently there is ε > 0 such that [u,s + ε) ⊂ U; but this contradicts the definition of least upper bound. It follows that s = b and [u,b) ⊂ U. Similarly, (a,u] ⊂ U which implies that X =(a,b) ⊂ U ⊂ X, andwe conclude that U = X. The reader may verify that this proof applies to the other types of interval (open, semi-open or infinite); alternatively, once every finite or infinite, open or closed interval is the telescopic union of a sequence of closed intervals, for instance, (0,1)= � n≥2 � � 1 1 ,1 − , n n the remainder of the proof follows from part 2 of Theorem (I.1.13). � 2 It is enough to consider the extrema of the interval a = infA and b = sup A, if they exist. If infA does not exist, set a = −∞;ifsupA does not exist, set b =+∞.
12 I Fundamental Concepts In particular, the real line R is connected. The previous theorem has a converse: (I.1.15) Theorem. Any connected subspace of R is an interval. Proof. If X ⊂ R is not an interval, we can find real numbers a,b ∈ X and c �∈ X such that a < c < b. In this case, U =(−∞,c) ∩ X and V = X ∩ (c,+∞) are non-empty (a ∈ U and b ∈ V), open in X, disjoint, and such that X = U ∪ V. Then X is not connected, contradicting the hypothesis. � The last two theorems have an immediate and interesting consequence: (I.1.16) Corollary. Let X be a connected space and f : X → R be a map. Then f (X) is an interval. (I.1.17) Remark. The concept of connectedness provides a simple method for establishing when two spaces are homeomorphic. The criterion goes as follows: suppose that f : X → Y is a homeomorphism; then, for every x ∈ X, the restriction of f to X � {x} is a homeomorphism from X � {x} onto Y � { f (x)}; therefore, X � {x} is connected if and only if Y � { f (x)} is connected. So, how does this method work? We wish to prove, for instance, that a semi-open interval (a,b] cannot be homeomorphic to an open interval (c,d). Suppose that a homeomorphism f : (a,b] → (c,d) could exist; then, we would have a homeomorphism (a,b)=(a,b] � {b} ∼ = (c,d) � { f (b)} which is impossible, for the first space is connected (it is an interval), but the second is not! Here is an useful result. (I.1.18) Theorem. Let X be a topological space and A one of its subspaces such that its closure in X coincides with X. Then, if A is connected, so is X. Proof. Let f : X →{0,1} be a two-valued map of X. Its restriction f |A is a twovalued map of A and, since A is connected, f |A is constant; we may suppose that f |A = 0 with no loss in generality. On the other hand, since A = X and f is continuous, we have f (A) ⊂ f (A)={0} = {0} and so f is constant in X. Therefore, X is connected. � (I.1.19) Remark. Let X be a topological space and x be one of its points; let {Xj, j ∈ J} be the set of all connected subspaces of X which contain x; then, by Theorem (I.1.13) (part 2), the union
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Page 11 and 12: x Preface same qualitative properti
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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
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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
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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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160 IV Cohomology groups Q and R ar
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162 IV Cohomology (IV.2.2) Corollar
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164 IV Cohomology Since g and π r
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166 IV Cohomology It is easily prov
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168 IV Cohomology ∩: B p (K;Z) ×
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Chapter V Triangulable Manifolds V.
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V.1 Topological Manifolds 173 is a
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V.1 Topological Manifolds 175 |Ki|
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V.2 Closed Surfaces 177 we define C
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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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