6
PETER HILTON
PROOF.
We may replace
X,
if necessary, by a homotopy-equivalent CW-com-
plex
Y
with just one vertex and no cells of positive dimension
r.
Then the
fat wedge Tn X has all the cells of xn of dimension
nr.
It follows that if dim
X
~
nr -
1 then the diagonal map .6: X
~
xn may be deformed into Tn X, so
that LS-catX
~
n.
If
we now choose n to be minimal subject to dim
X~
nr- 1,
we obtain the statement of the theorem. D
We now turn our attention to conditions under which there is no increase in
LS-cat in passing from
Y
to
C f,
where
f:
X
~
Y.
3. The quasi-homomorphism theorem
We start by recalling the Puppe sequence of the map
f:
X
~
Y. Thus we
simply start with
f:
X
~
Y;
construct the mapping cone
Z
=
C f,
with the
embedding k: Y
~
Z; and then construct the mapping cone of k. But, according
to
(2.4),
we may replace the associated embedding with the projection
Z
~EX.
It
then turns out that the next application of the mapping cone construction effectively
yields EX
~
EY; and from now on we have a periodic sequence
(3.1)
This Puppe sequence yields, for any space B, an exact sequence of pointed sets
(3.2)
k* f* t* k* f*
...
~
[EZ,B]
~
[EY,B]
~
[EX,B]
~
[Z,B]
~
[Y,B]
~
[X,B],
where [A, B] is the set of (pointed) homotopy classes of pointed maps from A to B;
note that [En A, B] admits a natural group structure, commutative if n
~
2.
Then
all but the last 3 sets in (3.2) are groups, and all but the last 3 functions in (3.2)
are homomorphisms. However, we can strengthen this last statement in a manner
crucial to what follows.
Let us pinch the cone C X in Z (
=
C f) at the level
t
= ~
to the base point.
This pinching map is a map
(3.3)
c:
z~Exvz,
where the range of
c
is the wedge or one-point union
4
of the suspension EX and
z.
Then c induces
c*:
[EXV Z,B]
~
[Z,B],
or
(3.4)
c*:
[EX, B] x
[Z,
B]
~
[Z, B],
which defines an operation of the group [EX, B] on the pointed set [Z, B]. (In
[EHl], we describe c as a cooperation of EX on Z.) We write gu for the map
u,g c from Z to B, for g: Z
~
B; u:
EX~
B, so that we may think of
gu as being obtained by operating on g with
u;
we would use a similar notation
when passing to homotopy classes. We also stress that the operation is natural (see
[EHl]).
4
The wedge is just the coproduct in the category of pointed spaces.
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