COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES
CONTENTS CHAPTER 1 AN INTRODUCTION TO THE HOMOTOPY GROUPS OF SPHERES 1 CHAPTER 2 SETTING UP THE ADAMS SPECTRAL SEQUENCE 41 CHAPTER 3 THE CLASSICAL ADAMS SPECTRAL SEQUENCE 59 CHAPTER 4 BP THEORY AND THE ADAMS NOVIKOV SPECTRAL SEQUENCE 102 CHAPTER 1 AN INTRODUCTION TO THE HOMOTOPY GROUPS OF SPHERES THIS CHAPTER IS INTENDED TO BE AN EXPOSITORY INTRODUCTION TO THE REST OF THE BOOK WE WILL INFORMALLY DESCRIBE THE SPECTRAL SEQUENCES OF ADAMS AND NOVIKOV WHICH ARE THE SUBJECT OF THE REMAINING CHAPTERS OUR AIM HERE IS TO GIVE A CONCEPTUAL PICTURE SUPPRESSING AS MANY TECHNICAL DETAILS AS POSSIBLE IN SECTION 1 WE LIST SOME THEOREMS WHICH ARE CLASSICAL IN THE SENSE THAT THEY DO NOT REQUIRE ANY OF THE MACHINERY DESCRIBED IN THIS BOOK THESE INCLUDE THE HUREWICZ THEOREM 1 1 2 THE FREUDENTHAL SUSPENSION THEOREM 1 1 4 THE SERRE NITE NESS THEOREM 1 1 8 THE NISHIDA NILPOTENCE THEOREM 1 1 9 AND THE COHEN MOORE NEISENDORFER EXPONENT THEOREM 1 1 10 THEY ALL PERTAIN DIRECTLY TO THE HOMOTOPY GROUPS OF SPHERES AND ARE NOT TREATED ELSEWHERE HERE THE HOMOTOPY GROUPS OF THE STABLE ORTHOGONAL GROUP SO ARE GIVEN BY THE BOTT PERIODICITY THEOREM 1 1 11 IN 1 1 12 WE DE NE THE J HOMOMORPHISM FROM I SO N TO N I SN ITS IMAGE IS GIVEN IN 1 1 13 AND IN 1 1 14 WE GIVE ITS COKERNEL IN LOW DIMENSIONS MOST OF THE FORMER IS PROVED IN SECTION 5 3 IN SECTION 2 WE DESCRIBE SERRE S METHOD OF COMPUTING HOMOTOPY GROUPS USING COHOMOLOGICAL TECHNIQUES IN PARTICULAR WE SHOW HOW TO ND THE RST ELEMENT OF ORDER P IN S3 1 2 4 THEN WE EXPLAIN HOW THESE METHODS WERE STREAMLINED BY ADAMS TO GIVE HIS CELEBRATED SPECTRAL SEQUENCE 1 2 10 THE NEXT FOUR THEOREMS DESCRIBE THE HOPF INVARIANT ONE PROBLEM A TABLE SHOWING THE ADAMS SPECTRAL SEQUENCE AT THE PRIME 2 THROUGH DIMENSION 45 IS GIVEN IN 1 2 15 IN CHAPTER 2 WE GIVE A MORE DETAILED ACCOUNT OF HOW THE SPECTRAL SEQUENCE IS SET UP INCLUDING A CONVERGENCE THEOREM IN CHAPTER 3 WE MAKE MANY CALCULATIONS WITH IT AT THE PRIME 2 IN 1 2 16 WE SUMMARIZE ADAMS S METHOD FOR PURPOSES OF COMPARING IT WITH THAT OF NOVIKOV THE BASIC IDEA IS TO USE COMPLEX COBORDISM 1 2 17 IN PLACE OF ORDINARY MOD P COHOMOLOGY FIG 1 2 19 IS A TABLE OF THE ADAMS NOVIKOV SPECTRAL SEQUENCE FOR COMPARISON WITH FIG 1 2 15 IN THE NEXT TWO SECTIONS WE DESCRIBE THE ALGEBRA SURROUNDING THE E2 TERM OF THE ADAMS NOVIKOV SPECTRAL SEQUENCE TO THIS END FORMAL GROUP LAWS ARE DE NED IN 1 3 1 AND A COMPLETE ACCOUNT OF THE RELEVANT THEORY IS GIVEN IN APPENDIX 2 THEIR CONNECTION WITH COMPLEX COBORDISM IS THE SUBJECT OF QUILLEN S THEOREM 1 3 4 AND IS DESCRIBED MORE FULLY IN SECTION 4 1 THE ADAMS NOVIKOV E2 TERM IS DESCRIBED IN TERMS OF FORMAL GROUP LAW THEORY 1 3 5 AND AS AN EXT GROUP OVER A CERTAIN HOPF ALGEBRA 1 3 6 THE REST OF SECTION 3 IS CONCERNED WITH THE GREEK LETTER CONSTRUCTION A METHOD OF PRODUCING IN NITE PERIODIC FAMILIES OF ELEMENTS IN THE E2 TERM AND IN FAVORABLE CASES IN THE STABLE HOMOTOPY GROUPS OF SPHERES THE BASIC DE NITIONS ARE GIVEN IN 1 1 3 17 AND 1 3 19 AND THE MAIN ALGEBRAIC FACT REQUIRED IS THE MORAVA LANDWEBER THEOREM 1 3 16 APPLICATIONS TO HOMOTOPY ARE GIVEN IN 1 3 11 1 3 15 AND 1 3 18 THE SECTION ENDS WITH A DISCUSSION OF THE PROOFS AND POSSIBLE EXTENSIONS OF THESE RESULTS THIS MATERIAL IS DISCUSSED MORE FULLY IN CHAPTER 5 IN SECTION 4 WE DESCRIBE THE DEEPER ALGEBRAIC PROPERTIES OF THE E2 TERM WE START BY INTRODUCING BP AND DE NING A HOPF ALGEBROID THE FORMER IS A MINIMAL WEDGE SUMMAND OF MU LOCALIZED AT A PRIME A HOPF ALGEBROID IS A GENERALIZED HOPF ALGEBRA NEEDED TO DESCRIBE THE ADAMS NOVIKOV E2 TERM MORE CONVENIENTLY IN TERMS OF BP 1 4 2 THE ALGEBRAIC AND HOMOLOGICAL PROPERTIES OF SUCH OBJECTS ARE THE SUBJECT OF APPENDIX 1 NEXT WE GIVE THE LAZARD CLASSI CATION THEOREM FOR FORMAL GROUP LAWS 1 4 3 OVER AN ALGEBRAICALLY CLOSED ELD OF CHARACTERISTIC P WHICH IS PROVED IN SECTION A2 2 THEN WE COME TO MORAVA S POINT OF VIEW THEOREM 1 3 5 DESCRIBES THE ADAMS NOVIKOV E2 TERM AS THE COHOMOLOGY OF A CERTAIN GROUP G WITH COE CIENTS IN A CERTAIN POLYNOMIAL RING L SPEC L IN THE SENSE OF ABSTRACT ALGEBRAIC GEOMETRY IS AN IN NITE DIMENSIONAL A NE SPACE ON WHICH G ACTS THE POINTS IN SPEC L CAN BE THOUGHT OF AS FORMAL GROUP LAWS AND THE G ORBITS AS ISOMORPHISM CLASSES AS DESCRIBED IN 1 4 3 THIS ORBIT STRUCTURE IS DESCRIBED IN 1 4 4 FOR EACH ORBIT THERE IS A STABILIZER OR ISOTROPY SUBGROUP OF G CALLED SN ITS COHOMOLOGY IS RELATED TO THAT OF G 1 4 5 AND ITS STRUCTURE IS KNOWN THE THEORY OF MORAVA STABILIZER ALGEBRAS IS THE ALGEBRAIC MACHINERY NEEDED TO EXPLOIT THIS FACT AND IS THE SUBJECT OF CHAPTER 6 OUR NEXT TOPIC THE CHROMATIC SPECTRAL SEQUENCE 1 4 8 THE SUBJECT OF CHAPTER 5 CONNECTS THE THEORY ABOVE TO THE ADAMS NOVIKOV E2 TERM THE GREEK LETTER CONSTRUCTION TS INTO THIS APPARATUS VERY NEATLY SECTION 5 IS ABOUT UNSTABLE HOMOTOPY GROUPS OF SPHERES AND IS NOT NEEDED FOR THE REST OF THE BOOK ITS INTRODUCTION IS SELF EXPLANATORY 1 CLASSICAL THEOREMS OLD AND NEW HOMOTOPY GROUPS THE HUREWICZ AND FREUDENTHAL THEOREMS STABLE STEMS THE HOPF MAP SERRE S NITENESS THEOREM NISHIDA S NILPOTENCE THEOREM COHEN MOORE AND NEISENDORFER S EXPONENT THEOREM BOTT PERIODICITY THE J HOMOMORPHISM WE BEGIN BY RECALLING SOME DE NITIONS THE NTH HOMOTOPY GROUP OF A CONNECTED SPACE X N X IS THE SET OF HOMOTOPY CLASSES OF MAPS FROM THE N SPHERE SN TO X THIS SET HAS A NATURAL GROUP STRUCTURE WHICH IS ABELIAN FOR N 2 WE NOW STATE THREE CLASSICAL THEOREMS ABOUT HOMOTOPY GROUPS OF SPHERES PROOFS CAN BE FOUND FOR EXAMPLE IN SPANIER 1 1 1 1 THEOREM 1 S1 Z AND M S1 0 FOR M 1 1 1 2 HUREWICZ S THEOREM N SN Z AND M SN 0 FOR M N A GENERATOR OF N SN IS THE CLASS OF THE IDENTITY MAP FOR THE NEXT THEOREM WE NEED TO DE NE THE SUSPENSION HOMOMORPHISM M SN M 1 SN 1 1 1 3 DEFINITION THE KTH SUSPENSION KX OF A SPACE X IS THE QUOTIENT OF IK X OBTAINED BY COLLAPSING IK X ONTO IK IK BEING THE BOUNDARY OF IK THE K DIMENSIONAL CUBE NOTE THAT I JX I JX AND KF KX KY IS THE QUOTIENT OF 1 F IK X IK Y IN PARTICULAR GIVEN F SM SN WE HAVE F SM 1 SN 1 WHICH INDUCES A HOMOMORPHISM M SN M 1 SN 1 2 1 1 4 FREUDENTHAL SUSPENSION THEOREM THE SUSPENSION HOMOMORPHISM N K SN N K 1 SN Q DE NED ABOVE IS AN ISOMORPHISM FOR K N 1 AND A SURJECTION FOR K N 1 1 1 5 COROLLARY THE GROUP N K SN DEPENDS ONLY ON K IF N K 1 1 1 6 DEFINITION THE STABLE K STEM OR KTH STABLE HOMOTOPY GROUP OF SPHERES S K IS N K SN FOR N K 1 THE GROUPS N K SN ARE CALLED STABLE IF N K 1 AND UNSTABLE IF N K 1 WHEN DISCUSSING STABLE GROUPS WE WILL NOT MAKE ANY NOTATIONAL DISTINCTION BETWEEN A MAP AND ITS SUSPENSIONS THE SUBSEQUENT CHAPTERS OF THIS BOOK WILL BE CONCERNED WITH MACHINERY FOR COMPUTING THE STABLE HOMOTOPY GROUPS OF SPHERES MOST OF THE TIME WE WILL NOT BE CONCERNED WITH UNSTABLE GROUPS THE GROUPS S K ARE KNOWN AT LEAST FOR K 45 SEE THE TABLES IN APPENDIX 3 ALONG WITH THEOREM 1 1 13 HERE IS A TABLE OF S K FOR K 15 K 0 1 2 3 4 5 6 7 8 S K Z Z 2 Z 2 Z 24 0 0 Z 2 Z 240 Z 2 2 K 9 10 11 12 13 14 15 S K Z 2 3 Z 6 Z 504 0 Z 3 Z 2 2 Z 480 Z 2 THIS SHOULD CONVINCE THE READER THAT THE GROUPS DO NOT FALL INTO ANY OBVIOUS PATTERN LATER IN THE BOOK HOWEVER WE WILL PRESENT EVIDENCE OF SOME DEEP PATTERNS NOT APPARENT IN SUCH A SMALL AMOUNT OF DATA THE NATURE OF THESE PATTERNS WILL BE DISCUSSED LATER IN THIS CHAPTER WHEN HOMOTOPY GROUPS WERE RST DE NED BY HUREWICZ IN 1935 IT WAS HOPED THAT N K SN 0 FOR K 0 SINCE THIS WAS ALREADY KNOWN TO BE THE CASE FOR N 1 1 1 1 THE RST COUNTEREXAMPLE IS WORTH EXAMINING IN SOME DETAIL 1 1 7 EXAMPLE 3 S2 Z GENERATED BY THE CLASS OF THE HOPF MAP S3 S2 DE NED AS FOLLOWS REGARD S2 AS RIEMANN DID AS THE COMPLEX NUMBERS C WITH A POINT AT IN NITY S3 IS BY DE NITION THE SET OF UNIT VECTORS IN R4 C2 HENCE A POINT IN S3 IS SPECI ED BY TWO COMPLEX COORDINATES Z1 Z2 DE NE BY Z1 Z2 Z1 Z2 IF Z2 0 IF Z2 0 IT IS EASY TO VERIFY THAT IS CONTINUOUS THE INVERSE IMAGE UNDER OF ANY POINT IN S2 IS A CIRCLE SPECI CALLY THE SET OF UNIT VECTORS IN A COMPLEX LINE THROUGH THE ORIGIN IN C2 THE SET OF ALL SUCH LINES BEING PARAMETERIZED BY S2 CLOSER EXAMINATION WILL SHOW THAT ANY TWO OF THESE CIRCLES IN S3 ARE LINKED ONE CAN USE QUATERNIONS AND CAYLEY NUMBERS IN SIMILAR WAYS TO OBTAIN MAPS S7 S4 AND S15 S8 RESPECTIVELY BOTH OF THESE REPRESENT GENERATORS OF IN NITE CYCLIC SUMMANDS THESE THREE MAPS AND WERE ALL DISCOVERED BY HOPF 1 AND ARE THEREFORE KNOWN AS THE HOPF MAPS WE WILL NOW STATE SOME OTHER GENERAL THEOREMS OF MORE RECENT VINTAGE 1 1 8 FINITENESS THEOREM SERRE N K SN IS NITE FOR K 0 EXCEPT WHEN N 2M K 2M 1 AND 4M 1 S Z FM WHERE FM IS NITE 3