Simplicial Structures in Topology

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References 1. J.W. Alexander – A proof of the invariance of certain constants of analysis situs, Trans.Am. Math. Soc. 16 (1915), 148–154. 2. J.W. Alexander – On the chains of a complex and their duals, Proc. Natl. Acad. Sci. USA 21 (1935), 509–511. 3. P.S. Aleksandrov – Poincaré and Topology, In: Proceedings of Symposia in Pure Mathematics, vol. 39, Part 2, American Mathematical Society, Providence (1983). 4. M. Barr – Acyclic Models, American Mathematical Society, Providence 2002. 5. E. Betti – Sopra gli spazi di un numero qualunque di dimensioni, Ann. Mat. Pura Appl. 4 (1871), 140–158. 6. E. Čech – Höherdimensionale Homotopiegruppen, In: Proceedings of the International Congress of Mathematicians. vol. 3, Zürich (1932). 7. A. Dold – Lectures on Algebraic Topology, Springer, New York 1972. 8. P.H. Doyle and D.A. Moran – A Short Proof that Compact 2-Manifolds Can Be Triangulated, Inventiones Math. 5 (1968), 160–168. 9. B. Eckmann and P. J. Hilton – Groupes d’homotopie et dualité, C. R. Acad. Sci. Paris Sér. A-B. 246 (1958), 2444–2447. 10. S. Eilenberg – Cohomology and Continuous Mappings, Ann. Math. 41 (1940), 231–251. 11. S. Eilenberg and S. MacLane – General Theory of Natural Equivalences, Trans.Am.Math. Soc. 58 (1945), 231–294. 12. S. Eilenberg and S. MacLane – Acyclic Models, Am. J. Math. 75 (1953), 189–199. 13. S. Eilenberg and N. Steenrod – Axiomatic Approach to Homology Theory, Proc. Natl. Acad. Sci. USA 31 (1945),117–120. 14. R. Fritsch and R.A. Piccinini – Cellular Structures in Topology, Cambridge University Press, Cambridge 1990. 15. K. Gruenberg – Introduzione all’Algebra Omologica, Pitagora Editrice, Bologna 2002. 16. P.J. Hilton and U. Stammbach – A Course in Homological Algebra, GTM 4, Springer, New York 1971. 17. P. Hilton and S. Wylie – Homology Theory, Cambridge University Press, Cambridge 1960. 18. H. Hopf – Über die Abbildungen der dreidinsionalen Sphäre auf die Kugelfläche, Math. Ann. 104 (1931), 637–665. 19. W. Hurewicz – Beiträge zur Topologie und Deformationen, Proc. Akad. Wetensch. Amsterdam 38 (1935), 112–119. 20. Sze-Tsen Hu – The Homotopy Addition Theorem, Ann. Math. 58 (1953), 108–122. 21. H. Künneth – Über die Bettische Zahlen einer Produktmannigfeltigkeit, Math. Ann. 91 (1923), 65–85. 22. S. Lefschetz – Algebraic Topology, Colloquium Publications, vol. XXVII, American Mathematical Society, New York 1942. 23. J.B. Listing – Vorstudiem zur Topologie, Gott. Stud. 1 (1847), 811–875. 239
240 References 24. C.R.F. Maunder – Algebraic Topology, Cambridge University Press, Cambridge 1970. 25. W.S. Massey – Algebraic Topology: an introduction, Harcourt, Brace & World, New York 1967. 26. R.A. Piccinini – Lectures on Homotopy Theory, Elsevier, North-Holland 1992. 27. H. Poincaré –Analysis Situs, J.École Polytechnique 1 (1895), 1–121. 28. H. Poincaré –Analyse de ses travaux scientifiques, Summary written in 1901 by H. Poincaré at the request of Mittag-Leffler, published in Acta Math. 38 (1921), 36–135. 29. T. Radó –Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged 2 (1925), 101–121. 30. H. Seifert and W. Threlfall – Lehrbuch der Topologie, Teubner Verlag, Stuttgart 1934. 31. J.-P. Serre – Groupes d’homotopie et classes de groupes abéliens, Ann. Math. 58 (1953), 258–294. 32. E. Spanier – Algebraic Topology, Springer, New York 1966. 33. G.W. Whitehead – Elements of Homotopy Theory, Springer, New York 1978. 34. H. Whitney – On Maps of An n-Sphere into Another n-Sphere, Duke Math. J. 3 (1937), 51–55. 35. H. Whitney – On Products in a Complex, Ann. Math. 39 (1938), 397–432.
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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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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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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
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Page 213 and 214: 196 VI Homotopy Groups and define t
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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 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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