A User's Guide to Algebraic TopologyWe have tried to design this book for both instructional and reference use, during and after a first course in algebraic topology aimed at users rather than developers; indeed, the book arose from such courses taught by the authors. We start gently, with numerous pictures to illustrate the fundamental ideas and constructions in homotopy theory that are needed in later chapters. A certain amount of redundancy is built in for the reader's convenience: we hope to minimize :fiipping back and forth, and we have provided some appendices for reference. The first three are concerned with background material in algebra, general topology, manifolds, geometry and bundles. Another gives tables of homo topy groups that should prove useful in computations, and the last outlines the use of a computer algebra package for exterior calculus. Our approach has been that whenever a construction from a proof is needed, we have explicitly noted and referenced this. In general, wehavenot given a proof unless it yields something useful for computations. As always, the only way to un derstand mathematics is to do it and use it. To encourage this, Ex denotes either an example or an exercise. The choice is usually up to you the reader, depending on the amount of work you wish to do; however, some are explicitly stated as ( unanswered) questions. In such cases, our implicit claim is that you will greatly benefit from at least thinking about how to answer them. |
Contents
III | 5 |
IV | 8 |
VI | 9 |
VII | 10 |
VIII | 17 |
IX | 27 |
X | 37 |
XI | 44 |
LX | 227 |
LXI | 230 |
LXII | 233 |
LXIII | 234 |
LXIV | 243 |
LXV | 244 |
LXVII | 245 |
LXVIII | 246 |
XII | 45 |
XIII | 47 |
XIV | 51 |
XV | 53 |
XVI | 56 |
XVII | 58 |
XVIII | 62 |
XIX | 67 |
XX | 69 |
XXI | 76 |
XXII | 80 |
XXIII | 86 |
XXIV | 89 |
XXV | 93 |
XXVII | 96 |
XXVIII | 99 |
XXIX | 105 |
XXX | 106 |
XXXI | 108 |
XXXII | 116 |
XXXIII | 118 |
XXXIV | 122 |
XXXV | 125 |
XXXVI | 133 |
XXXVII | 135 |
XXXVIII | 143 |
XXXIX | 144 |
XL | 147 |
XLI | 150 |
XLII | 151 |
XLIII | 155 |
XLIV | 157 |
XLV | 164 |
XLVI | 165 |
XLVIII | 167 |
XLIX | 168 |
L | 172 |
LI | 175 |
LII | 181 |
LIII | 189 |
LIV | 204 |
LV | 209 |
LVI | 210 |
LVII | 219 |
LVIII | 222 |
LIX | 224 |
LXIX | 248 |
LXXI | 250 |
LXXII | 257 |
LXXIII | 260 |
LXXIV | 267 |
LXXV | 271 |
LXXVI | 277 |
LXXVII | 283 |
LXXX | 285 |
LXXXI | 286 |
LXXXII | 288 |
LXXXV | 289 |
LXXXVII | 291 |
LXXXVIII | 292 |
LXXXIX | 295 |
XC | 297 |
XCII | 298 |
XCIII | 300 |
XCIV | 301 |
XCV | 305 |
XCVI | 309 |
XCVII | 310 |
XCVIII | 315 |
XCIX | 318 |
C | 319 |
CI | 321 |
CII | 322 |
CIII | 331 |
CIV | 332 |
CV | 335 |
CVII | 348 |
CVIII | 349 |
CIX | 351 |
CX | 352 |
CXI | 354 |
CXII | 362 |
CXIV | 365 |
CXV | 366 |
CXVI | 368 |
CXVIII | 369 |
CXIX | 370 |
CXX | 381 |
385 | |
393 | |
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Common terms and phrases
abelian groups antipodal barycentric subdivision bijection BO(n boundary called cells chain complex closed cochain coefficients cofibration cofunctor cohomology theory commutative compact connected construction continuous map Corollary CW-complex define Definition denote diagram differential dimension exact sequence example exists extension fiber bundle fibration finite fixed point functor G-bundle geometry given GL(n group G H¹(X Hausdorff space Hence homeomorphism homology and cohomology homotopy classes homotopy equivalence homotopy groups inclusion induces isomorphisms ISBN isomorphism Lemma linear manifold map f map ƒ mapping cylinder metric morphism n-connected obtain orientable pair principal bundle problem Proof quotient retract simplicial approximation simplicial complex spectral sequence spheres structure subcomplex subgroup subset subspace surjection Switzer 106 tangent tensor Theorem topological space Topp torus triangulation trivial vector field vector space