| Preface |
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vii | |
| 1 Singular homology |
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1 | (52) |
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1 Introduction: singular simplices and chains |
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1 | (5) |
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6 | (4) |
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3 Categories, functors, and natural transformations |
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10 | (3) |
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13 | (3) |
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5 Homotopy, star-shaped regions |
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16 | (6) |
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6 Homotopy invariance of homology |
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22 | (2) |
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24 | (3) |
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27 | (4) |
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9 Homology long exact sequence |
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31 | (5) |
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10 Excision and applications |
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36 | (6) |
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11 Eilenberg-Steenrod axioms and the locality principle |
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42 | (4) |
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46 | (3) |
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13 Proof of the locality principle |
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49 | (4) |
| 2 Computational methods |
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53 | (50) |
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57 | (5) |
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16 Homology of CW complexes |
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62 | (4) |
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66 | (2) |
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18 Euler characteristic and homology approximation |
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68 | (4) |
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72 | (2) |
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74 | (7) |
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81 | (3) |
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22 Fundamental theorem of homological algebra |
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84 | (4) |
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88 | (6) |
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24 Universal coefficient theorem |
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94 | (2) |
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25 Kiinneth and Eilenberg-Zilber |
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96 | (7) |
| 3 Cohomology and duality |
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103 | (58) |
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26 Coproducts, cohomology |
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103 | (5) |
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108 | (4) |
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28 Products in cohomology |
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112 | (3) |
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29 Cup product, continued |
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115 | (4) |
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30 Surfaces and nondegenerate symmetric bilinear forms |
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119 | (5) |
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31 Local coefficients and orientations |
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124 | (8) |
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32 Proof of the orientation theorem |
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132 | (4) |
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33 A plethora of products |
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136 | (4) |
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34 Cap product and Cech cohomology |
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140 | (5) |
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35 Cech cohomology as a cohomology theory |
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145 | (4) |
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36 Fully relative cap product |
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149 | (3) |
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152 | (4) |
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156 | (5) |
| 4 Basic homotopy theory |
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161 | (44) |
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39 Limits, colimits, and adjunctions |
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161 | (5) |
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40 Cartesian closure and compactly generated spaces |
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166 | (5) |
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41 Basepoints and the homotopy category |
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171 | (5) |
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176 | (3) |
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43 Fibrations, fundamental groupoid |
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179 | (6) |
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185 | (4) |
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45 Cofibration sequences and co-exactness |
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189 | (4) |
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46 Weak equivalences and Whitehead's theorems |
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193 | (5) |
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47 Homotopy long exact sequence and homotopy fibers |
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198 | (7) |
| 5 The homotopy theory of CW complexes |
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205 | (28) |
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48 Serre fibrations and relative lifting |
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205 | (4) |
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49 Connectivity and approximation |
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209 | (4) |
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213 | (4) |
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51 Hurewicz, Eilenberg, Mac Lane, and Whitehead |
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217 | (5) |
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52 Representability of cohomology |
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222 | (4) |
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226 | (7) |
| 6 Vector bundles and principal bundles |
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233 | (28) |
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233 | (5) |
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55 Principal bundles, associated bundles |
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238 | (4) |
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56 G-CW complexes and the I-invariance of BunG |
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242 | (4) |
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57 The classifying space of a group |
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246 | (4) |
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58 Simplicial sets and classifying spaces |
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250 | (5) |
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59 The Cech category and classifying maps |
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255 | (6) |
| 7 Spectral sequences and Serre classes |
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261 | (66) |
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60 Why spectral sequences? |
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261 | (4) |
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61 Spectral sequence of a filtered complex |
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265 | (6) |
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62 Serre spectral sequence |
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271 | (5) |
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276 | (6) |
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64 Gysin sequence, edge homomorphisms, and transgression |
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282 | (7) |
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65 Serre exact sequence and the Hurewicz theorem |
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289 | (6) |
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66 Double complexes and the Dress spectral sequence |
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295 | (5) |
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67 Cohomological spectral sequences |
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300 | (7) |
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307 | (6) |
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69 Mod C Hurewicz and Whitehead theorems |
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313 | (5) |
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70 Freudenthal, James, and Bousfield |
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318 | (9) |
| 8 Characteristic classes, Steenrod operations, and cobordism |
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327 | (56) |
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71 Chern classes, Stiefel-Whitney classes, and the Leray-Hirsch theorem |
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327 | (8) |
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72 H (BU (n)) and the splitting principle |
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335 | (5) |
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73 Thom class and Whitney sum formula |
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340 | (7) |
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74 Closing the Chern circle, and Pontryagin classes |
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347 | (6) |
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353 | (8) |
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361 | (8) |
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369 | (5) |
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78 Applications of cobordism |
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374 | (9) |
| Bibliography |
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383 | (4) |
| Index |
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387 | |
