LUSTERNIK-SCHNIRELMANN CATEGORY IN HOMOTOPY THEORY
13
It is to be expected that our remark in Section 5 that there is a weak version of
category, and that n-fold cup-products of positive-dimensional cohomology classes
vanish in a space X with wcat X
~
n, should have a valid dual. This turns out to be
the case. Let
F X
be the homotopy fiber of
j :
XV
X
----+
X
x
X
with associated map
k:
FX----+
XV
X.
We then say that
X
is a weak H-space if Vk
~
0:
FX----+ X.
Obviously, an H-space is a weak H-space. Obviously, too, Whitehead products
vanish in a weak H -space, since we immediately infer from
i"l
~
0 that "'
~
ke,
for some
e:
sm+n-1-
F(m,n), the homotopy fiber of
j:
sm
v
sn- sm
X
sn.
Then the diagram
F(m,n)
(6.4)
e /
1
k
sm+n-1
~
smvsn
shows8 that [a,
.B]
=
0 in X.
F(a,{3)
-
aV{3
-
FX
lk
xvx
\.0
V'
-
X
We also have a dual of Theorem 3.2. Thus if Y is an H-space with structure
map
J.Ly: Y
x
Y----+ Y,
we describe
f: Y----+ X
as a
quasihomomorphism (qh)
if
there is a map
J.L: X
x
X
----+
X
such that the diagram
YxY
l
J.LY
y
!..!J
XxX
lJ.L
X
(homotopy-)commutes. Let Z be the homotopy-fiber of/, with associated map
k: Z----+Y.
Then
THEOREM 6.2.
If f:
Y
----+
X
is a qh, then
Z
is an
H
-space and may be
furnished with a structure map J.Lz : Z
x
Z
----+
Z such that k: Z
----+
Y is a
homomorphism.
PROOF. We consider the diagram
nx
l
z k y f
X
-
-
-
j /-LY
iJ.L
ZxZ
kxk
YxY
fxf
XxX
-
----+
i iz
i
jy
zvz
kVk
YVY
-
Then
JJ.Ly(k
x
k)
~
J.L(/k
x
fk)
~
0, so there exists
v: Z
x
Z
----+
Z
with
kv
~
J.Ly(k
x
k).
Moreover,
kvjz
~
J.Ly(k
x
k)jz
=
J.Lyjy(k V k)
=
\7y(k V k)
=
k\7 z.
Now, by the dual of Theorem 3.1, there is an action of the group [A, nX] on the
set
[A,
Z], natural in
A,
such that, for any h1, h2:
A----+ Z,
(6.5)
kh1
~
kh2: A
----+
Y {::} there exists
u: A
----+
nx
with
h2
~
hf.
8
We leave the reader the task of finding a weak H-space which is not an H-space.
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