A CONCISE COURSE IN ALGEBRAIC TOPOLOGY
CONTENTS INTRODUCTION CHAPTER 1 THE FUNDAMENTAL GROUP AND SOME OF ITS APPLICATIONS 1 WHAT IS ALGEBRAIC TOPOLOGY? 2 THE FUNDAMENTAL GROUP 3 DEPENDENCE ON THE BASEPOINT 4 HOMOTOPY INVARIANCE 5 CALCULATIONS: ?1(R) = 0 AND ?1(S1) = Z 6 THE BROUWER ?XED POINT THEOREM 7 THE FUNDAMENTAL THEOREM OF ALGEBRA CHAPTER 2 CATEGORICAL LANGUAGE AND THE VAN KAMPEN THEOREM 1 CATEGORIES 2 FUNCTORS 3 NATURAL TRANSFORMATIONS 4 HOMOTOPY CATEGORIES AND HOMOTOPY EQUIVALENCES 5 THE FUNDAMENTAL GROUPOID 6 LIMITS AND COLIMITS 7 THE VAN KAMPEN THEOREM 8 EXAMPLES OF THE VAN KAMPEN THEOREM CHAPTER 3 COVERING SPACES 1 THE DE?NITION OF COVERING SPACES 2 THE UNIQUE PATH LIFTING PROPERTY 3 COVERINGS OF GROUPOIDS 4 GROUP ACTIONS AND ORBIT CATEGORIES 5 THE CLASSI?CATION OF COVERINGS OF GROUPOIDS 6 THE CONSTRUCTION OF COVERINGS OF GROUPOIDS 7 THE CLASSI?CATION OF COVERINGS OF SPACES 8 THE CONSTRUCTION OF COVERINGS OF SPACES CHAPTER 4 GRAPHS 1 THE DE?NITION OF GRAPHS 2 EDGE PATHS AND TREES 3 THE HOMOTOPY TYPES OF GRAPHS 4 COVERS OF GRAPHS AND EULER CHARACTERISTICS 5 APPLICATIONS TO GROUPS CHAPTER 5 COMPACTLY GENERATED SPACES 1 THE DE?NITION OF COMPACTLY GENERATED SPACES 2 THE CATEGORY OF COMPACTLY GENERATED SPACES CHAPTER 6 CO?BRATIONS 1 THE DE?NITION OF CO?BRATIONS 2 MAPPING CYLINDERS AND CO?BRATIONS 3 REPLACING MAPS BY CO?BRATIONS 4 A CRITERION FOR A MAP TO BE A CO?BRATION 5 CO?BER HOMOTOPY EQUIVALENCE CHAPTER 7 FIBRATIONS 1 THE DE?NITION OF ?BRATIONS 2 PATH LIFTING FUNCTIONS AND ?BRATIONS 3 REPLACING MAPS BY ?BRATIONS 4 A CRITERION FOR A MAP TO BE A ?BRATION 5 FIBER HOMOTOPY EQUIVALENCE 6 CHANGE OF ?BER CHAPTER 8 BASED CO?BER AND ?BER SEQUENCES 1 BASED HOMOTOPY CLASSES OF MAPS 2 CONES, SUSPENSIONS, PATHS, LOOPS 3 BASED CO?BRATIONS 4 CO?BER SEQUENCES 5 BASED ?BRATIONS 6 FIBER SEQUENCES 7 CONNECTIONS BETWEEN CO?BER AND ?BER SEQUENCES CHAPTER 9 HIGHER HOMOTOPY GROUPS 1 THE DE?NITION OF HOMOTOPY GROUPS 2 LONG EXACT SEQUENCES ASSOCIATED TO PAIRS 3 LONG EXACT SEQUENCES ASSOCIATED TO ?BRATIONS 4 A FEW CALCULATIONS 5 CHANGE OF BASEPOINT 6 N-EQUIVALENCES, WEAK EQUIVALENCES, AND A TECHNICAL LEMMA CHAPTER 10 CW COMPLEXES 1 THE DE?NITION AND SOME EXAMPLES OF CW COMPLEXES 2 SOME CONSTRUCTIONS ON CW COMPLEXES 3 HELP AND THE WHITEHEAD THEOREM 4 THE CELLULAR APPROXIMATION THEOREM 5 APPROXIMATION OF SPACES BY CW COMPLEXES 6 APPROXIMATION OF PAIRS BY CW PAIRS 7 APPROXIMATION OF EXCISIVE TRIADS BY CW TRIADS CHAPTER 11 THE HOMOTOPY EXCISION AND SUSPENSION THEOREMS 1 STATEMENT OF THE HOMOTOPY EXCISION THEOREM 2 THE FREUDENTHAL SUSPENSION THEOREM 3 PROOF OF THE HOMOTOPY EXCISION THEOREM CHAPTER 12 A LITTLE HOMOLOGICAL ALGEBRA 1 CHAIN COMPLEXES 2 MAPS AND HOMOTOPIES OF MAPS OF CHAIN COMPLEXES 3 TENSOR PRODUCTS OF CHAIN COMPLEXES 4 SHORT AND LONG EXACT SEQUENCES CHAPTER 13 AXIOMATIC AND CELLULAR HOMOLOGY THEORY 1 AXIOMS FOR HOMOLOGY 2 CELLULAR HOMOLOGY 3 VERI?CATION OF THE AXIOMS 4 THE CELLULAR CHAINS OF PRODUCTS 5 SOME EXAMPLES: T, K, AND RP N CHAPTER 14 DERIVATIONS OF PROPERTIES FROM THE AXIOMS 1 REDUCED HOMOLOGY; BASED VERSUS UNBASED SPACES 2 CO?BRATIONS AND THE HOMOLOGY OF PAIRS 3 SUSPENSION AND THE LONG EXACT SEQUENCE OF PAIRS 4 AXIOMS FOR REDUCED HOMOLOGY 5 MAYER-VIETORIS SEQUENCES 6 THE HOMOLOGY OF COLIMITS CHAPTER 15 THE HUREWICZ AND UNIQUENESS THEOREMS 1 THE HUREWICZ THEOREM 2 THE UNIQUENESS OF THE HOMOLOGY OF CW COMPLEXES CHAPTER 16 SINGULAR HOMOLOGY THEORY 1 THE SINGULAR CHAIN COMPLEX 2 GEOMETRIC REALIZATION 3 PROOFS OF THE THEOREMS 4 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY 5 CLASSIFYING SPACES AND K(?, N)S CHAPTER 17 SOME MORE HOMOLOGICAL ALGEBRA 1 UNIVERSAL COE?CIENTS IN HOMOLOGY 2 THE K¨UNNETH THEOREM 3 HOM FUNCTORS AND UNIVERSAL COE?CIENTS IN COHOMOLOGY 4 PROOF OF THE UNIVERSAL COE?CIENT THEOREM 5 RELATIONS BETWEEN ?AND HOM CHAPTER 18 AXIOMATIC AND CELLULAR COHOMOLOGY THEORY 1 AXIOMS FOR COHOMOLOGY 2 CELLULAR AND SINGULAR COHOMOLOGY 3 CUP PRODUCTS IN COHOMOLOGY 4 AN EXAMPLE: RP N AND THE BORSUK-ULAM THEOREM 5 OBSTRUCTION THEORY CHAPTER 19 DERIVATIONS OF PROPERTIES FROM THE AXIOMS 1 REDUCED COHOMOLOGY GROUPS AND THEIR PROPERTIES 2 AXIOMS FOR REDUCED COHOMOLOGY 3 MAYER-VIETORIS SEQUENCES IN COHOMOLOGY 4 LIM1 AND THE COHOMOLOGY OF COLIMITS 5 THE UNIQUENESS OF THE COHOMOLOGY OF CW COMPLEXES CHAPTER 20 THE POINCAR´E DUALITY THEOREM 1 STATEMENT OF THE THEOREM 2 THE DE?NITION OF THE CAP PRODUCT 3 ORIENTATIONS AND FUNDAMENTAL CLASSES 4 THE PROOF OF THE VANISHING THEOREM 5 THE PROOF OF THE POINCAR´E DUALITY THEOREM 6 THE ORIENTATION COVER CHAPTER 21 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY 1 THE EULER CHARACTERISTIC OF COMPACT MANIFOLDS 2 THE INDEX OF COMPACT ORIENTED MANIFOLDS 3 MANIFOLDS WITH BOUNDARY 4 POINCAR´E DUALITY FOR MANIFOLDS WITH BOUNDARY 5 THE INDEX OF MANIFOLDS THAT ARE BOUNDARIES CHAPTER 22 HOMOLOGY, COHOMOLOGY, AND K(?, N)S 1 K(?, N)S AND HOMOLOGY 2 K(?, N)S AND COHOMOLOGY 3 CUP AND CAP PRODUCTS 4 POSTNIKOV SYSTEMS 5 COHOMOLOGY OPERATIONS CHAPTER 23 CHARACTERISTIC CLASSES OF VECTOR BUNDLES 1 THE CLASSI?CATION OF VECTOR BUNDLES 2 CHARACTERISTIC CLASSES FOR VECTOR BUNDLES 3 STIEFEL-WHITNEY CLASSES OF MANIFOLDS 4 CHARACTERISTIC NUMBERS OF MANIFOLDS 5 THOM SPACES AND THE THOM ISOMORPHISM THEOREM 6 THE CONSTRUCTION OF THE STIEFEL-WHITNEY CLASSES 7 CHERN, PONTRYAGIN, AND EULER CLASSES 8 A GLIMPSE AT THE GENERAL THEORY CHAPTER 24 AN INTRODUCTION TO K-THEORY 1 THE DE?NITION OF K-THEORY 2 THE BOTT PERIODICITY THEOREM 3 THE SPLITTING PRINCIPLE AND THE THOM ISOMORPHISM 4 THE CHERN CHARACTER; ALMOST COMPLEX STRUCTURES ON SPHERES 5 THE ADAMS OPERATIONS 6 THE HOPF INVARIANT ONE PROBLEM AND ITS APPLICATIONS CHAPTER 25 AN INTRODUCTION TO COBORDISM 1 THE COBORDISM GROUPS OF SMOOTH CLOSED MANIFOLDS 2 SKETCH PROOF THAT N?IS ISOMORPHIC TO ??(TO) 3 PRESPECTRA AND THE ALGEBRA H?(TO; Z2) 4 THE STEENROD ALGEBRA AND ITS COACTION ON H?(TO) 5 THE RELATIONSHIP TO STIEFEL-WHITNEY NUMBERS 6 SPECTRA AND THE COMPUTATION OF ??(TO) = ??(MO) 7 AN INTRODUCTION TO THE STABLE CATEGORY SUGGESTIONS FOR FURTHER READING 1 A CLASSIC BOOK AND HISTORICAL REFERENCES 2 TEXTBOOKS IN ALGEBRAIC TOPOLOGY AND HOMOTOPY THEORY 3 BOOKS ON CW COMPLEXES 4 DI?ERENTIAL FORMS AND MORSE THEORY 5 EQUIVARIANT ALGEBRAIC TOPOLOGY 6 CATEGORY THEORY AND HOMOLOGICAL ALGEBRA 7 SIMPLICIAL SETS IN ALGEBRAIC TOPOLOGY 8 THE SERRE SPECTRAL SEQUENCE AND SERRE CLASS THEORY 9 THE EILENBERG-MOORE SPECTRAL SEQUENCE 10 COHOMOLOGY OPERATIONS 11 VECTOR BUNDLES 12 CHARACTERISTIC CLASSES 13 K-THEORY 14 HOPF ALGEBRAS; THE STEENROD ALGEBRA, ADAMS SPECTRAL SEQUENCE 15 COBORDISM 16 GENERALIZED HOMOLOGY THEORY AND STABLE HOMOTOPY THEORY 17 QUILLEN MODEL CATEGORIES 18 LOCALIZATION AND COMPLETION; RATIONAL HOMOTOPY THEORY 19 IN?NITE LOOP SPACE THEORY 20 COMPLEX COBORDISM AND STABLE HOMOTOPY THEORY 21 FOLLOW-UPS TO THIS BOOK