LUSTERNIK-SCHNIRELMANN CATEGORY IN HOMOTOPY THEORY 7

We now have the following important refinement to the statement of the ex-

actness of (3.2).

THEOREM

3.1.

Given

h1, h2:

Z----+ B, then

h 1k

~

h2 k: Y----+

B

=

there exists

u:

EX----+ B with

h2 ~

h't..

We may now state the main result of this section. Suppose that LS-cat Y

~

n,

with

n

~

2, and structure map

f..Ly: Y----+ TnY.

We then say that

f: X----+

Yin

(3.1) is a

quasi-homomorphism

if there exists

w

X

----+

rn X

such that

X

!J.L

rnx

y

!

f..LY

rny

homotopy commutes; of course if LS-cat

X

~

n

with structure map

f..L,

then

f

is a

homomorphism. We may now prove

THEOREM

3.2.

Iff in

(3.1)

is a quasi-homomorphism then LS-cat Z

~

n and

Z admits a structure-map f..LZ such that k is a homomorphism.

PROOF.

Consider the diagram

X

f

y

k

z

l

EX ----+ ----+ ----+

J.L! f..LY!

rnx

Tnf

rny

Tnk

rnz

----+

----+

!jy

!

iz

yn

kn

zn

----+

Then, since

rn k

0

f..LY

0

f

~

rn k

0

rn

f

0

f..L

~

0, it follows that there exists

v : z

----+

rn z

such that

vk

~

rn k

0

f..LY.

But then

jzvk

~

jz

o

Tnk

o

f..LY

=

knjyJ.Ly

~

knD.y

=

D.zk.

Thus, by Theorem 3.1, there exists

g:

EX ----+

zn

with

D.z

~

(jzv)Y.

We now

argue that

j z

* maps [EX,

rn

Z] onto [EX,

zn];

it is at this point that we require

n

~

2. For

jz.

is a group-homomorphism and [EX,

zn]

is the direct product of n

copies of [EX, Z]. Let

ti

embed Z as the ith factor in

zn,

so that

n

(3.5)

[Ex,zn]

=II

ti.[EX,Z].

i=l

It then suffices to show that

ti*

[EX, Z]

~

j

z

*[EX,

rn

Z]. But this is clear, since

there is an embedding

/'i,i

of

z

in

rn

z

such that

j z

/'i,i

=

£i.

Thus we may choose

h:

EX ----+

rn Z

so that

jzh

~

g.

Then, by naturality,

D.z ~ (izv)izh ~ iz(vh).

We set

vh

=

f..LZ: Z----+ TnZ.

Then

D.z

~

izf..Lz,

so that

f..Lz

is a structure map for

LS-cat Z

~

n. Finally, by another application of Theorem 3.1,

J.Lzk

=

vhk

~

vk

~

Tnkof..Ly,

so that

k: Y

----+Z is a homomorphism, as claimed.

D