ALGEBRAIC TOPOLOGY

By G. NAVIY

CHAPTER 0 SOME UNDERLYING GEOMETRIC NOTIONS HOMOTOPY AND HOMOTOPY TYPE 1 CELL COMPLEXES 5 OPERATIONS ON SPACES 8 TWO CRITERIA FOR HOMOTOPY EQUIVALENCE 10 THE HOMOTOPY EXTENSION PROPERTY 14 CHAPTER 1 THE FUNDAMENTAL GROUP 1 1 BASIC CONSTRUCTIONS PATHS AND HOMOTOPY 25 THE FUNDAMENTAL GROUP OF THE CIRCLE 29 INDUCED HOMOMORPHISMS 34 1 2 VAN KAMPEN’S THEOREM FREE PRODUCTS OF GROUPS 41 THE VAN KAMPEN THEOREM 43 APPLICATIONS TO CELL COMPLEXES 50 1 3 COVERING SPACES LIFTING PROPERTIES 60 THE CLASSI?CATION OF COVERING SPACES 63 DECK TRANSFORMATIONS AND GROUP ACTIONS 70 ADDITIONAL TOPICS 1 A GRAPHS AND FREE GROUPS 83 1 B K(G,1) SPACES AND GRAPHS OF GROUPS 87 CHAPTER 2 HOMOLOGY 2 1 SIMPLICIAL AND SINGULAR HOMOLOGY ?COMPLEXES 102 SIMPLICIAL HOMOLOGY 104 SINGULAR HOMOLOGY 108 HOMOTOPY INVARIANCE 110 EXACT SEQUENCES AND EXCISION 113 THE EQUIVALENCE OF SIMPLICIAL AND SINGULAR HOMOLOGY 128 2 2 COMPUTATIONS AND APPLICATIONS DEGREE 134 CELLULAR HOMOLOGY 137 MAYER-VIETORIS SEQUENCES 149 HOMOLOGY WITH COEF?CIENTS 153 2 3 THE FORMAL VIEWPOINT AXIOMS FOR HOMOLOGY 160 CATEGORIES AND FUNCTORS 162 ADDITIONAL TOPICS 2 A HOMOLOGY AND FUNDAMENTAL GROUP 166 2 B CLASSICAL APPLICATIONS 169 2 C SIMPLICIAL APPROXIMATION 177 CHAPTER 3 COHOMOLOGY 3 1 COHOMOLOGY GROUPS THE UNIVERSAL COEF?CIENT THEOREM 190 COHOMOLOGY OF SPACES 197 3 2 CUP PRODUCT THE COHOMOLOGY RING 211 A K¨UNNETH FORMULA 218 SPACES WITH POLYNOMIAL COHOMOLOGY 224 3 3 POINCAR´E DUALITY ORIENTATIONS AND HOMOLOGY 233 THE DUALITY THEOREM 239 CONNECTION WITH CUP PRODUCT 249 OTHER FORMS OF DUALITY 252 ADDITIONAL TOPICS 3 A UNIVERSAL COEF?CIENTS FOR HOMOLOGY 261 3 B THE GENERAL K¨UNNETH FORMULA 268 3 C H–SPACES AND HOPF ALGEBRAS 281 3 D THE COHOMOLOGY OF SO(N) 292 3 E BOCKSTEIN HOMOMORPHISMS 303 3 F LIMITS AND EXT 311 3 G TRANSFER HOMOMORPHISMS 321 3 H LOCAL COEF?CIENTS 327 CHAPTER 4 HOMOTOPY THEORY 4 1 HOMOTOPY GROUPS DE?NITIONS AND BASIC CONSTRUCTIONS 340 WHITEHEAD’S THEOREM 346 CELLULAR APPROXIMATION 348 CW APPROXIMATION 352 4 2 ELEMENTARY METHODS OF CALCULATION EXCISION FOR HOMOTOPY GROUPS 360 THE HUREWICZ THEOREM 366 FIBER BUNDLES 375 STABLE HOMOTOPY GROUPS 384 4 3 CONNECTIONS WITH COHOMOLOGY