14 PETER HILTON
Thus there exists g:
zv
z
~
nx
with
'\lz
~
(vjz)9.
But
jz:
[Z
X
z,nx]
~
[Z
v
z,
OX] is a surjective homomorphism, so g
~
hjz
for some h:
z
X
z
~
nx'
and, if
f..LZ = vh,
then, by naturality,
'\lz ~ (vjz)hiz ~ J.Lzjz,
showing that Z is an H-space with structure map
f..LZ·
Finally, using
(6.5)
again,
kf..Lz = k(vh) ~ kv ~ f..Ly(k
x
k),
so that
k
is a homomorphism. D
In conclusion, we state one final result dual to a result on suspensions; namely,
there are H-spaces which are not loop-spaces. The most easily accessible example
is 8
7
.
References
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[BH1] I. Berstein and P. J. Hilton, Category and generalized Hopf invariants, Ill. Journ. Math.
4 (1960), 437- 451.
[BH2] I. Berstein and P. J. Hilton, Suspensions and comultiplications, Topology 2 (1963), 63-
82.
[EH1] B. Eckmann and P. J. Hilton, Operators and co-operators in homotopy theory, Math.
Annalen 141 (1960), 1 - 21.
[EH2] B. Eckmann and P. J. Hilton, Grouplike structures in general categories; III Primitive
categories, Math. Annalen 150 (1963), 165- 187.
[Fo] R. H. Fox, On the Lustemik-Schnirelmann category, Ann. of Math. 42 (1941), 333- 370.
[Ga] T. Ganea, Lustemik-Schnirelmann category and cocategory, Proc. London Math. Soc. 10
(3) (1960), 623 - 634.
[Hi] P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30
(1955), 154 - 172.
[HMR] P. J. Hilton, G. Mislin and J. Roitberg, On co-H-spaces, Comm. Math. Helv. 53 (1978),
1-14.
[Ja] I. M. James, On category in the sense of Lustemik-Schnirelmann, Topology 17 (1978),
331-348.
[LS] L. Lusternik and L. Schnirelmann, Methodes Topologiques dans les Problemes Variation-
nels, Hermann, Paris (1934).
[Wh] G. W. Whitehead, The homology suspension, Colloque de topologie algebrique tenue
aLouvain (1956), 89- 95.
DEPARTMENT OF MATHEMATICAL SCIENCES, STATE UNIVERSITY OF NEW YORK, BINGHAM-
TON, NEW YORK 13902-6000 USA AND DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CENTRAL
FLORIDA, ORLANDO, FLORIDA 32816-1364, USA
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