A USER'S GUIDE TO SPECTRAL SEQUENCES
1 AN INFORMAL INTRODUCTION 1 1 “THERE IS A SPECTRAL SEQUENCE ” 1 2 LACUNARY PHENOMENA 1 3 EXPLOITING FURTHER STRUCTURE 1 4 WORKING BACKWARDS 1 5 INTERPRETING THE ANSWER 2 WHAT IS A SPECTRAL SEQUENCE? 2 1 DE?NITIONS AND BASIC PROPERTIES 2 2 HOW DOES A SPECTRAL SEQUENCE ARISE? 2 3 SPECTRAL SEQUENCES OF ALGEBRAS 2 4 ALGEBRAIC APPLICATIONS 3 CONVERGENCE OF SPECTRAL SEQUENCES 3 1 ON CONVERGENCE 3 2 LIMITS AND COLIMITS 3 3 ZEEMAN’S COMPARISON THEOREM PART II: TOPOLOGY 4 TOPOLOGICAL BACKGROUND 4 1 CW-COMPLEXES 4 2 SIMPLICIAL SETS 4 3 FIBRATIONS 4 4 HOPF ALGEBRAS AND THE STEENROD ALGEBRA 5 THE LERAY-SERRE SPECTRAL SEQUENCE I 5 1 CONSTRUCTION OF THE SPECTRAL SEQUENCE 5 2 IMMEDIATE APPLICATIONS 5 3 APPENDICES 6 THE LERAY-SERRE SPECTRAL SEQUENCE II 6 2 THE TRANSGRESSION 6 3 CLASSIFYING SPACES AND CHARACTERISTIC CLASSES 6 4 OTHER CONSTRUCTIONS OF THE SPECTRAL SEQUENCE ? N 7 THE EILENBERG-MOORE SPECTRAL SEQUENCE I 7 1 DIFFERENTIAL HOMOLOGICAL ALGEBRA 7 2 BRINGING IN THE TOPOLOGY 7 3 THE KOSZUL COMPLEX 7 4 THE HOMOLOGY OF QUOTIENT SPACES OF GROUP ACTIONS 8 THE EILENBERG-MOORE SPECTRAL SEQUENCE II 8 1 ON HOMOGENEOUS SPACES 8 2 DIFFERENTIALS IN THE EILENBERG-MOORE SPECTRAL SEQUENCE 8 3 FURTHER STRUCTURE 8BIS NONTRIVIAL FUNDAMENTAL GROUPS 8BIS 1 ACTIONS OF THE FUNDAMENTAL GROUP 8BIS 2 HOMOLOGY OF GROUPS 8BIS 3 NILPOTENT SPACES AND GROUPS 9 THE ADAMS SPECTRAL SEQUENCE 9 1 MOTIVATION: WHAT COHOMOLOGY SEES 9 2 MORE HOMOLOGICAL ALGEBRA; THE FUNCTOR EXT 9 3 THE SPECTRAL SEQUENCE 9 4 OTHER GEOMETRIC APPLICATIONS 9 5 COMPUTATIONS 9 6 FURTHER STRUCTURE 10 THE BOCKSTEIN SPECTRAL SEQUENCE 10 1 THE BOCKSTEIN SPECTRAL SEQUENCE 10 2 OTHER BOCKSTEIN SPECTRAL SEQUENCES PART III: SINS OF OMISSION 11 MORE SPECTRAL SEQUENCES IN TOPOLOGY 11 1 SPECTRAL SEQUENCES FOR MAPPINGS AND SPACES OF MAPPINGS 11 2 SPECTRAL SEQUENCES AND SPECTRA 11 3 OTHER ADAMS SPECTRAL SEQUENCES 11 4 EQUIVARIANT MATTERS 11 5 MISCELLANEA 12 SPECTRAL SEQUENCES IN ALGEBRA, GEOMETRY AND ANALYSIS 12 1 SPECTRAL SEQUENCES FOR RINGS AND MODULES 12 2 SPECTRAL SEQUENCES IN GEOMETRY 12 3 SPECTRAL SEQUENCES IN ALGEBRAIC K-THEORY 12 4 DERIVED CATEGORIES