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.