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
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