Contemporary Mathematics
Volume 12, 1982
JOSE ADEM'S CONTRIBUTION TO ALGEBRAIC TOPOLOGY
Samuel Gitler
It gives me great pleasure to have all of you here, contributing
with so many good lectures and with your company to celebrate Jose's six-
tieth birthday.
I will try to give you a chronological outline of Adem's work.
Suppose X is a finite CW-complex and R a ring. Let
Hq(X; R) denote the qth cohomology group of X with coefficients
v
in R. Alexander, Cech and Whitney around 1938 defined a product pairing,
the cup product:
making H*(X; R)
=
{HP(x; R)} a graded commutative algebra; i.e., satisfying
xy
=
(-l)dim x dim Yyx.
Hopf had introduced in 1935 an integer invariant for a mapping
f:
s2q-l
--- Sq, the Hopf invariant. Steenrod gave the following inter-
pretation of it. Form the mapping cone X of the mapping f; then
x
=
sq u
e2q
f
so that Hq(X)
=
H
2q(X)
=
Z, and if u and v are corresponding
generators, then
u2
=
AV.
The integer
A
is the Hopf invariant of f.
Hopf had shown the existence of maps of Hopf invariant one for
q
=
1, 2, 4 and 8. He also showed that
x3(s2)
=
Z. Freudenthal
obtained his suspension theorem in 1937, which in particular gave
n
xn+l(S) # 0.
Steenrod asked himself how could he prove cohomologically the es-
sentiality of a map sn+l --- sn, suspension of the Hopf map f:
s3
---
s2.
Following his reformulation of the Hopf invariant, if
X=
s2
u
e4
,
f
1
© 1982 American Mathematical Society
0271-4132/81/0000-0153/$02.75
http://dx.doi.org/10.1090/conm/012/01
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