PREFACE
At th e en d o f th e secon d Worl d War , th e subjec t o f homotop y theor y wa s i n a ver y
rudimentary state . Sinc e the n it s developmen t ha s bee n ver y rapid , an d i t ha s ha d a
profound effec t o n othe r branche s o f topology . I n fact , th e reductio n o f geometri c
questions t o problem s i n homotop y theor y i s almos t "standar d operatin g procedure*' .
Examples o f thi s approac h ar e th e classificatio n o f fibr e bundle s an d th e determina -
tion o f th e rin g o f cobordis m classe s o f manifolds .
Early i n th e histor y o f th e subjec t i t wa s observe d tha t man y phenomen a becam e
much mor e regula r i n th e * "stable range" . Thi s ha s le d t o a particularl y rapi d
development o f stabl e homotop y theor y a s a subjec t i n it s ow n right . Yet , despit e al l
the grea t advance s tha t hav e bee n made , suc h i s ou r ignoranc e tha t ther e i s no t a
single stabl y nontrivia l finit e comple x whos e stabl e homotop y group s ar e completel y
known.
These lecture s ar e intende d t o describ e som e o f thes e advances , particularl y i n
the stabl e theory . Becaus e o f th e breadt h o f th e subject , I hav e mad e n o attemp t a t
completeness, bu t hav e chose n instea d t o discus s thos e topic s whic h hav e bee n
closest t o m y ow n interests .
Chapter I i s devote d t o genera l homolog y an d cohomolog y theories , pairings , an d
duality theorems . I n Chapte r II , semi-sirnplicia l spectr a ar e introduce d a s a settin g
for m y join t wor k wit h Ka n o n Poincar e duality . Chapte r II I discusse s th e Adam s
spectral sequenc e an d som e o f it s recen t unstabl e analogues . Chapte r I V i s devote d
to stabl e homotop y group s a s module s ove r th e stabl e homotop y rin g o* , an d Chapte r
V t o som e o f th e modul e extension s whic h aris e naturall y i n th e subject .
These lecture s wer e delivere d a t th e 1969—197 0 Holida y Mathematic s
Symposium a t th e Ne w Mexic o Stat e Universit y a t La s Cruces , a s par t o f th e
Regional Conferenc e Progra m sponsore d b y th e Conferenc e Boar d o f th e Mathematica l
Sciences wit h th e suppor t o f th e Nationa l Scienc e Foundation , I wis h t o expres s m y
appreciation t o th e Conferenc e Boar d fo r makin g the m possible , an d t o th e
Department o f Mathematica l Science s o f th e Universit y fo r thei r invitation , a s wel l a s
for thei r lavis h hospitalit y durin g th e conference .
George W , Whitehea d
Massachusetts Institut e o f Technolog y
June 16, 1970
IV
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