10 PETER HILTON
We now consider
x
2l
=
X x X/ XV X. A simple application of Van Kampen's
Theorem shows us that
1r1X( 2)
=
{1 },
x2l
is !-connected. On the other hand,
an application of the Kiinneth formula shows us that
x2l
has no non-vanishing
homology groups in positive dimensions. We conclude that
x2l
is contractible, so
that certainly
q.6.
~ 0: X
---+
x2l
and so wcat X
=
2 (note that X is not itself
contractible).
On the other hand, Theorem 4.2 shows us that it is false that cat
X
=
2, since
1r1X is not free. We know by Corollary 2.5 that cat
X
=
3 or 4.
5.2. The strong version.
In the strong version, first suggested by Tudor
Ganea, we say that the strong category of
X
is less than or equal to n, that is,
scat
X
S
n
if
X
has the homotopy type of a CW-complex expressible as the union
of
n
(closed) contractible subcomplexes. Obviously, cat
X
S
n
if scat
X
S
n.
To show that strong category differs in general from category, it suffices to
consider the case
n
=
2. For it is plain that scat
X
=
2 if and only if
X
has the
homotopy type of a suspension space. Thus we will first show that, if X and Y
are suspensions, there are homomorphisms
f:
X
---+
Y which are not suspension
maps.
6
Now it was proved in
[BHl]
that a map
f:
sm
---+
sn
is a homomorphism
if and only if all the (generalized) Hopf invariants of
J,
in the sense of
[Hi],
vanish.
Now, if p is an odd prime, we know that the smallest stem in which an element
a
of order
p
occurs is the (2p - 3) stem and that such an element then occurs
for the first time in 1r2p(S3
).
Thus we have a:
S
2P
---+
S
3
of order p, and all its
Hopf invariants are 0, since they map
a
into finite groups
1r2p(Sn), n
odd,
n
2:
5,
containing no element of order
p.
Thus
a
is a homomorphism; however, it cannot
be a suspension element since
7r2p-
1
(S2
)
is a finite group, isomorphic to
7r2p_
1
(S3
),
which does not contain an element of order
p.
It is now clear that, if we use
a
to attach a (2p
+
1 )-cell to
S
3
,
then
S
3 Ua
e2P+l
is a space of category 2 which is not a suspension; for it is shown in
[BH2]
that
sm
Ua
en is a suspension if, and only if,
a
is a suspension element.
6. Dual notions
In [Ga], Ganea produced an inductive definition of LS-category, and was there-
by enabled to invent the dual notion of cocategory. Although this notion has sat-
isfactory general properties - for example, n-fold iterated Whitehead products
vanish in a space X such that cocat X
S
n - it has proved very difficult to do
calculations involving cocategory.
We will briefly describe Ganea's definitions. First he characterized LS-category
as follows. He constructed the diagram
6
What we are here calling homomorphisms were referred to in
[EH2]
as primitive maps.
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