Chapter I . Homolog y an d cohomolog y theorie s
It wa s i n 1945 tha t Eilenber g an d Steenro d [32] , [33 ] firs t announce d thei r cele =
brated axiom s fo r homolog y theory . O f thei r axioms , th e firs t si x ar e o f a ver y genera l
character; thei r seventh , th e dimension axiom, i s muc h mor e special . Tha t i t wa s
given equa l weigh t wit h th e other s i s undoubtedl y du e t o th e fac t that , a t tha t time ,
essentially n o interestin g example s wer e know n excep t fo r th e standar d ones .
It wa s i n 1953 tha t Spanie r an d J . H . C . Whitehea d [63]—[66 ] introduce d th e
suspension categor y a s a framewor k withi n whic h t o stud y stabl e phenomena . Whil e
homotopy an d cohomotop y group s d o no t satisf y th e axioms , th e correspondin g stabl e
groups d o (excep t fo r th e dimensio n axiom) . Thes e wer e th e firs t example s o f
"extraordinary" theories .
The nex t exampl e wa s du e t o Atiya h an d Hirzebruc h [12], [13]? wh o inaugurate d
the stud y o f stabl e equivalenc e classe s o f vecto r bundles . Thu s the y obtaine d th e
new cohomolog y theorie s ^
R
an d ^
c
. Thes e hav e ha d spectacula r succes s i n th e
solution o f som e o f th e classica l problem s o f topology—notabl y i n Adam' s solutio n o f
the vecto r fiel d proble m [3] .
About th e sam e tim e Atiya h [ l l ] an d Conne r an d Floy d [26 ] independentl y intro -
duced bordis m groups . O f al l homolog y theories , thes e ar e perhap s closes t t o th e
intuition. Tha t the y diffe r greatl y fro m th e classica l group s i s evince d b y th e fac t
that th e homolog y group s o f a point , fa r fro m satisfyin g th e dimensio n axiom , ar e jus t
Thorn's cobordis m group s [69] .
By thi s tim e i t wa s widel y realize d tha t cohomology-lik e functor s coul d b e
described i n term s o f mapping s int o som e fixe d '^universa l space" . E . H . Brow n [22 ]
succeeded i n givin g a n axiomati c descriptio n o f suc h functors . Th e descriptio n o f
cohomology theorie s i n term s o f map s int o spectr a the n followed .
General homolog y theorie s wer e studie d b y th e presen t autho r [72] . Probabl y th e
deepest result s o f th e classica l theor y ar e thos e whic h involv e homolog y an d cohomolog y
groups simultaneously ; fo r example
7
dualit y theorems . I t wa s show n tha t thes e carr y
over t o th e mor e genera l setting . I n particular , Alexande r duality , togethe r wit h
Brown's representatio n theorem , implie s tha t ther e i s essentiall y a one-to-on e corre -
spondence betwee n th e tw o kind s o f theories .
Less successfu l ha s bee n th e effor t t o exten d th e universa l coefficien t an d
Kunneth theorem s t o th e genera l setting . I t i s mor e o r les s clea r tha t th e shor t exac t
sequences o f th e classica l theor y mus t b e replace d b y spectra l sequences , bu t th e
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