THE ALGEBRA OF SECONDARY COHOMOLOGY OPERATIONS

By BARBARA MARTINEZ

CONTENTS I SECONDARY COHOMOLOGY AND TRACK CALCULUS UNSTABLE ALGEBRAS POWER ALGEBRAS CARTAN FORMULA ADEM RELATION THE THEORY OF EILENBERG-MACLANE SPACES THE EILENBERG-MACLANE SPACES ZN APPENDIX TO SECTION 2 1: SMALL MODELS OF EILENBERG-MACLANE SPACES GROUPOIDS OF MAPS TRACK CATEGORIES AND TRACK THEORIES SECONDARY COHOMOLOGY OPERATIONS THE SECONDARY STEENROD ALGEBRA THE STABLE TRACK THEORY OF EILENBERG-MACLANE SPACES STABLE SECONDARY COHOMOLOGY OPERATIONS MAPS AND TRACKS UNDER AND OVER A SPACE THE PARTIAL LOOP OPERATION THE PARTIAL LOOP FUNCTOR FOR EILENBERG-MACLANE SPACES NATURAL SYSTEMS TRACK EXTENSIONS COHOMOLOGY OF CATEGORIES SECONDARY COHOMOLOGY AND THE OBSTRUCTION OF BLANC SECONDARY COHOMOLOGY AS A STABLE MODEL WEAK ADDITIVE TRACK EXTENSIONS LINEARITY TRACKS THE ?-STRUCTURE OF THE SECONDARY STEENROD ALGEBRA THE COCYCLE OF [[KSTABLE P ]] THE KRISTENSEN DERIVATION APPENDIX TO SECTION 4 5: COMPUTATION OF ?[P] FOR P ODD OBSTRUCTION TO LINEARITY OF COCYCLES TRACK ALGEBRAS, PAIR ALGEBRAS AND CROSSED ALGEBRAS THE ?-PSEUDO FUNCTOR THE STRICTI?CATION OF A ?-TRACK ALGEBRA THE STRICTI?CATION OF A ?-TRACK MODULE THE STRICTI?CATION OF THE SECONDARY STEENROD ALGEBRA THE STRICTI?CATION OF SECONDARY COHOMOLOGY AND KRISTENSEN OPERATIONS TWO-STAGE OPERATION ALGEBRAS II PRODUCTS AND POWER MAPS IN SECONDARY COHOMOLOGY PERMUTATION ALGEBRAS SECONDARY PERMUTATION ALGEBRAS SECONDARY COHOMOLOGY AS A SECONDARY PERMUTATION ALGEBRA INDUCED HOMOTOPIES SQUARING MAPS SECONDARY COHOMOLOGY OF A PRODUCT SPACE THE BOREL CONSTRUCTION COMPARISON MAPS COMPARISON TRACKS POWER MAPS LINEARITY TRACKS FOR POWER MAPS CARTAN TRACKS FOR POWER MAPS ADEM TRACKS FOR POWER MAPS COHOMOLOGY AS A POWER ALGEBRA BOCKSTEIN TRACKS FOR POWER MAPS A LIST OF SECONDARY RELATIONS SECONDARY LINEARITY RELATIONS RELATIONS FOR ITERATED LINEARITY TRACKS PERMUTATION RELATIONS SECONDARY CARTAN RELATIONS CARTAN LINEARITY RELATION K¨UNNETH TRACKS K¨UNNETH-STEENROD OPERATIONS LINEARITY TRACKS FOR K¨UNNETH-STEENROD OPERATIONS CARTAN TRACKS FOR K¨UNNETH-STEENROD OPERATIONS THE INTERCHANGE RELATION FOR CARTAN TRACKS THE ASSOCIATIVITY RELATION FOR CARTAN TRACKS THE LINEARITY RELATION FOR CARTAN TRACKS STABLE K¨UNNETH-STEENROD OPERATIONS THE HOPF-ALGEBRA TG(EA) ?-TRACKS LINEARITY TRACKS ?? AND ??,? SUM AND PRODUCT OF ?-TRACKS THE ALGEBRA T?OF ?-TRACKS THE ALGEBRA OF LINEAR ?-TRACKS GENERALIZED CARTAN TRACKS AND THE ASSOCIATIVITY RELATION STABILITY OF CARTAN TRACKS THE RELATION DIAGONAL THE RIGHT ACTION ON THE RELATION DIAGONAL THE MONOIDAL CATEGORY OF [P]-ALGEBRAS THE SECONDARY DIAGONAL THE RIGHT ACTION ON THE SECONDARY DIAGONAL THE SECONDARY HOPF ALGEBRA B PAIR ALGEBRAS OVER THE SECONDARY HOPF ALGEBRA B SECONDARY COHOMOLOGY AS A PAIR ALGEBRA OVER B SECONDARY INSTABILITY THE OPERATORS S AND L THE EXTENDED LEFT ACTION OPERATOR THE INTERCHANGE ACTING ON SECONDARY COHOMOLOGY COMPUTATION OF THE EXTENDED LEFT ACTION COMPUTATION OF THE EXTENDED SYMMETRY THE TRACK FUNCTOR H?[ ] THE ?-CLASS OF B COMPUTATION OF THE ?-CLASS THE MULTIPLICATION CLASS OF B PROOF OF THE UNIQUENESS THEOREM RIGHT EQUIVARIANT COCYCLE OF B RIGHT EQUIVARIANT SPLITTING OF B COMPUTATION OF ? AND THE DIAGONAL ?1 OF B THE MULTIPLICATION IN B COMPUTATION OF THE MULTIPLICATION MAP ADMISSIBLE RELATIONS COMPUTATION OF B BIBLIOGRAPHY