LUSTERNIK-SCHNIRELMANN CATEGORY IN HOMOTOPY THEORY 11
Kn+l Fn+l
Un+l
X
----t ----t
i
II
Kn Fn
Un
X
----t ----t
i
II
i
II
K3 F3
U3
X
----t ----t
i
II
K2 F2
U2 X
----t ----t
i
II
nx
----t
*
----t
X
One starts with the map
*
-----
X and constructs the (homotopy- )fiber OX
-----

One then constructs the (homotopy-)cofiber
*
-----
F2 of this map and the canonical
extension u
2 :
F2
-----
X of
*
-----
X (note that F2
=
L:OX). The inductive step
is achieved by taking the fiber Kn
-----
Fn of Un and the cofiber Fn
-----
Fn+l of
this map; and then constructing the canonical extension Un+l: Fn+l -----X of Un.
Ganea then proved that
cat X :::;
n
¢:?
there exists v: X
-----
Fn with Un
v
~
ld.
The dual construction is now clear. The diagram is
I
X
un+l
F~+l K~+l
----t ----t
II
!
I
X
un
F'
K'
----t ----t n
n
II
!
...
II
!
I
X
u3
F'
K'
----t
3
----t
3
II
!
I
X
u2 F'
K'
----t
2
----t
2
II
!
X
----t
*
----t
L:X
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