Pub
.
Ma
.
UAB
Vol
.
30
ns
2-3
Des
.
1986
Le
n
be
a
ini e
g oup
and
le s
Bn
be
i s
classi ying
space
.
Wi h
e e y
subg oup

yc n

he e
is
associa ed
a
co e ing
i(y,n)
:
By
-
"
Bn
.
I
g
E
n
hen
mul iplica ionby
g
on
En

indu-
ces
a
map
cg
:
By
-
B(g
-1
yg)
.
Le
E
be
an
in ini eloop
space
.
Then
he e
is
he
ollowing
exac
sequence
i*,j*
(*)
0 -
[Bw
;E]
-~
II[Bn
;E]

II

[B(n
ngn
g-1)
;E]
nP

n
p
~gER
P P
P
whe e
u

is a
p-Sylow
subg oup
o n
,
p oduc s

II
...
and
P

n
nP

g lc
II . . .
a e
o e
all
p-Sylow
subg oups
o all

p

p imes
p
,
i *
=

II

i(i

ngn
P9
-1
;
3c )

and

j
*
=

II

i(gn
-1

gnn
"
n
)
oc

.
(see
(2])
.
nP

g

P

P

n
P
g

P

P
'

P

9
F om
he
sequence
(*)
i
ollows
ha
a
map
om
Bn
o
an
in ini e
loop
space
is
homo opic
o
ze o
i
and
only
i
i s
es ic-
ions
o
classi ying
spaces
o
all
Sylow
subg oups
a e
homo opic
o
ze o
.
We
wan
o
see
whe he
he
same
s a emen
is
ue
o
an
a bi-
a ysimply
connec ed
space
.
Fo
example
i
n =
IIx
p
hen
weha e
he
P
ollowing
p oposi ion
MAPS
FROM Bn
INTO
X
Zdzislaw
Woj kowiak
P oposi ion
1
.
I
X
is
simply-connec ed
hen
P oo
.
The map
VBn
p
- Bn
is a
homological
equi alence
.
The e o e
p
using
an
obs uc ion
heo y
we
ob ain
a
equi ed
isomo phism
o
any
simply
connec ed
space
X
.
In
u he
conside a ions
we
es ic
ou
a en ion o
a
e y
smallclass
o
g oups
.
Le
np
be
a
maximal
p-Sylowsubg oup
o n
Le
N(n
p
)

be
a
no malizo '
o

np in u
and
le
W
p =
N(np)/np
De ini ion
1
.
We
say
ha
n
sa is ies
W -condi ion
i
he
map
p
H*(n
;Z(P))

H*(i
;Z(P))
p

is
an
isomo phism
.
Examples
1
.
I
n

is a
no maldi iso
in
n
hen
W
-condi ion
is
sa is ied
.
p

p
2
.
I
n

is
abelian
hen
W
-condi ion
is
sa is ied
.
p

p
3
.
W
p
-condi ion
is
sa is ied
o he
bina yicosahed alg oup
I*
4
.
I
n
=
GL(n
;F
q
)

hen
W
p
-condi ion
is
sa is ied
o
some
No a ion
.
-

means
"is
homo opic
o"
.
p-Sylow
subg oup
o
n
.)
90
and
allp imesp
.
p imesp
[Ba
:X]
~z
n[Bn
p
;X]
p
Weha e
he
ollowing
sequence
o
co ib a ions

(up is a
maximal
i

j

6

S
(i)
(**)

Bnp
-

Bn

--~

Cone
(i)

= C

-
S
(Bn
p
)
1
S
(Bn)

-~
" "
Le

()
(P)

deno es
he
p-comple ion
unc o
and le

( )
(P)
deno es
he
p-localiza ion
unc o
.
A e
applying
(
)(P)
o
(**)
we
ob ain
he
ollowing
sequence
o
co ib a ions
(BnP)
(P)

=
BnP

P, (B
n)
(P)

p
_
C
(P)

- C
(P)

P
-
S(BnP)
(P)
=
S(Bwp)
S
(i)
P
.
S(Bn)
(P)
=
S(Bn)
(P)
-"
Fu he we
shall
deal
onlywi h
a
case
o
a
ixed
p íme p
and
he e-
o e
we
always
d op
he
índex
p in
iP,jP,6P,
. . . .
Theo em
1
.

(F
.
Cohen
[1
))

I

n

sa is ies

Wp
condi ion
hen
s
(i)

:
S(B P
)

-
S(Bn)
(P)

has
a
le
in e se

k

,
b k
:
C(P)
S(B7 )
(P)
-
S(Bi
P
)

is a
homo opy
equi alence
and
j

:

(B n)
P
-
C
(P)

is
homo op
ic

o
ze o
.
P oo
.
Le

k = jWP1
.
E e y
elemen

g
EW
P

induces
a
map
h
g
:
Bic
P
- Bn
P

(conjuga ion
by
g
) .
Le
N
=
%
S(h9)
:
S(Bi P
)
-
S(B, p)
g sW
and
le
k-N
= k
"
id-N
:
S(Bn
P
)
-
S(Bn
P
) .
One

P
easilychecks
ha
he
na u al
map
=
1
+ 2
:
S(Bn
)
P

Tel(N)
Tel(k-N)

is
a
homo opy
equi alence
.
E e y
elemen

g
e
WP
induces
also
a
map
hg
:
Bn
"
Bi
homo opic o
he
iden i y
.
Le

N
=

S
S(h9)
_
geW
P
=
k
:
(SBn)
(P)
-"
(SBn)
(P)
.
The
maps
Q:
Tel
(N)

-
Tel
(N)

and

-
,
:

(SBn)
(P)

-
Tel
(N)

a e
homo opy
equi a-
lences
.
Le

i
1
:
Tel(N)
-
Tel(N)
Tel(k-N)

be
he
na u al
inclusion
.
One can
check' ha

k =
.(
2+
1)-1o
i1oe-1' 1

is a
le
in e se
o
S(i)

.
The e o e

b
k

is a
homo opy
equi alence
.
I
es s
o
show
ha

j
-O
.
b

has
a
igh
in e se

.
This
implies
ha

j
-j
,
bo
.
Hence
weha e
ha

j
-0
.
Co olla y
1
.
I

n

sa is ies

W
P
-condi ion
and

X

is
simply-connec ed
and
p-local
hen
he
map
:
Bn
- X
is
homo opically
i ial
i
and
only
i
i s
es ic ion
o
B7
p
is
homo opically
i ial
.
P oo
.
I

.¡
-0

hen
. he e
is

'
:
C
- X

such ha

'oj
_
.
This
implies
ha

-O
.
(7
unde s and
i s
es ic ions
o
Tel(N) and
Tel(k-N)
.
L
emma
1
.
Le
us
suppose
ha
X
=
gY
.
Then
he e
is an
isomoüphism
P oo
.
Weha e
a
di ec sys em
o
spaces
The e
is
he
ollowing
exac
sequence
o
Milno
Le
us
no ice
ha
N
.N = k
"
N
( esp
.
(k-N)a(k-N)
=
k(k-N))

implies
ha
ou
in e se
sys ems
sa is y
he
Mi ag-Le le
condi ions
.
This
implies
ha
lim
1
e ms anish
.
0
I

e
[SBn
p
;S?Y]

and

Y

is
p-local
hen
o
any

n
e
Z
(P)

we
can
de ine
n .
in
he
ollowing
wo
ways
.
i)

Maps
.(S
1
;Y)
=
S2Y

has
he
same
hcmo opy
ype
as
Maps
.(S(
P)
;Y)
Fo
any
ne
Z
(p)

he e
is a
map_
n
:
S
1
-
S1
o
deg ee
n
and
we
de ine
n .
as
a
composi ion
no
.
Le
us
suppose
ha
we
ha e
a
map
:
SBn
P
-
X
.
We
wan o
[Tel(N) ( esp
.
Tel(k-N))
;X]
11

lim
[SBn
p ;X]
N( esp
.
k-N)
SBn
N( esp
.
k-N)
P
SB
n

-
.
P
O
-
"

lim
1

[SBn

,

X]

-
[Tel(N) ( esp
.k-N))
;X]

-

lim

[SBn
P
;X]-
O
N( esp
.
k-N)
P

N( esp
.
k-N)
ii)

Slh
Bn

S1P)
^ B
:
p

.,The
map

n

:

S
~P)

-
"
S
~P)

induces
P
n
:
S~
p)
^
B
.p

S(
P)
n
B
p
.
We
de ine
n
"
as
a
composi ion
.
.
Le

:
SBn
p
-
"
X
=
4Y
.
Le
us
se

i
=
k
"
( aN)

and

=
1
"
( -
(k-N))

.
Then

*

}

s
lim[SB7
;X]

and
2

k

1

kn
1
ne{1,2
. .
.} N

p
* =
{
1
}

e
lim[SBg
;X]
.
The e o e
by
Lemma
1
*
and
2

kn
2
ne{1,2
. .
.}
k-N

p

1
2

de ine
maps

*
:
Tel
(N)

- X

and

2
:
Tel
(k-N)

- X

.

*
*
es ic ed
o

SBn
P
(i
.e
.

( i
2)
o

whe e
=
i
+
2
:
SBn
p
-
"
Tel(N)
Tel(k N)

is a
sum
o
inclusions
on o
he
i s
segmen s
o
he
mapping
elescopes)
is
homo opic
o
k

-N

+-'

o
(k-N)

=

P opos
i ion
1
.
The
na u al
isomo phism
*
:
lim[SB7
;X]®lim[SBw
;X]
-
"
[SBwp
;X]
ÑP
k-N
P
is
gi en
by
(( n)
;(gn))
-
1+
9
1
.
The
in e se
map
is
gi en
by
-

( *~
; 2)
P oo
.
Themap
:
SBn - Tel(N)
Tel(k-N)

induces
a
map
P
[Tel(N)
;X]
®
[Tel(k-N)
;X]
-
"
[SBn
p
;X]

which
is
gi en
by
he sum
o
es ic ions
o
he
i s
segmen s
o
he
elescopes
.
This
shows
he
i s
pa
o
he
p oposi ion
.
By
he
p e ious
discussions
-
( *, 2)

de ines
a
map
in
h
e
opposi e
di ec ion
which
is
he
in e se
o *
.
Co olla y
2
.
I
-N
is
homo opic
o
k
"
hen
iii)

o

any

gE
W
P

we
ha e

ha

oS
(h
9
)
-

P oo
.
i)

ollows
om
he
de ini ion
o
E*
.
We
ha e eha
ob
^,
( 1 V
2)
o eó
^
.
io ob
- *
.P
-1.Qo
l
on

ioe
-lo
L
oS(i)
oil
^
.O
ii)
implies
ha
he e
is

l
:
SBR
- X

such
ha

'
.S(i)
-
.
This
implies
ha

oS
(h
g
)
-
-
El
Co olla y_
3
.
I

X
= 0
2
Y

and

X

is
simply
connec ed
hen
i
:
Bn - Bn
induces
an
isomo phism
p
W
L(B) )
p
;X]
_
[Bnp
;X] p
.
.
P oo
.
Weha e
ha
.z,oS(i)-
o
1

.
l

and

Q

a e
homo opy
equi a-
lences
.
The e o e
i is
enough
o
show ha
W
*
:
[Tel(N)
;X]
=
lim[SBi
;QY]
-+
[SBn
p
;QY]
p
NW
is
an
isomo phism
.
Le
us
suppose
ha

e[SBn
P
;4Y]
P

.
Then
*

_
{
ñ
1

k

.N

lim[SBn
P=
;oY]
and

*( 1)

.
This
}
n,{1,2,
...
}
e

-
implies
ha
*
is
an
epimo phism
.
*
is
also
a
monomo phism
and
he e o =_
i is
an
isomo phism
.
Theo em
2
.
I
X is a
nilpo en ,
p-local
space
and
i
n
sa is ies
W
p
-condi ion
hen
he
na u al
map
W
[Bn
;X]
-»
[Bn
p ;X] P
is
a
su jec ion
.
I
X
is a
loop
space
hen
W
[Bn
;X]

[Bn
p ;X]
P
is a
bijec ion
.
oo
.
We
ha e
al eadyp o ed
heo em

when X
is a
double
loop
space
.
Le
us
suppose
ha
X
is
a
loop
and
ha
X has
only
a
ini e
numbe
o non- i ialhomo opy
g oups
.
Le
us
conside
a
pa
o
he
Pos niko
owe
o
X
,
-
.
S?X
n-
,
L

"
-

K(
nn,n)

-1
X
n

c

Xn-1

d

K(ic
n
n+1)
Le
us
suppose
ha
he
heo em
is
ue
o
Xn-l
.
We ha e
he
ollow-
ing
commu a i e
diag am
[Bic,4X
n-1
1
:
[B
;K(Rn
,n)
1
b
[Bu
;X
n
1
-S-
[Bn,Xn-11
d
[B
.
jc
;K(7c
n
'n+1)
1
114
i

1ijj

1
k

i¡
Q

11
;
m
w
1

W
b

W
c
i

w
d
1
W
[BnP
,ox
n-1
1

p

[Bi
P
;K(n
n
'n)
1

P
-+
[Bn
P
;Xn1

P
y
[B c
P;Xn-1

P
1
P_
.
[Bic
;K(jc
n
;n+1)
]
P
we mus
show
ha
k
is a
bijec ion
.
I
k(x)
=
k(y)

hen
c(x)

=
c(y)

.
Hence
he e
exis s

zE[Bu
;K(n
n
;n)]

such ha

z
=
x
-1
.
y
.
This

implies
ha
.

1
.
3
(z)

=
k(x)

k
(y)

.
The e o e
he e

is
w
E
[B3

;
4X
n-1
]

such

ha

al
(w)

=
j
(z)

.

Le

w1=
k

E

woh
g

.

Then
gEw
p
W
a
l
(w
1
)
=
j
(z)

and

w
1
E
[Bnp ;QXn-11 P
.
The e
is

E
[Bn
;QX
n-1
1
such
ha

i
( )

=
w
1

.

we
ha e

j
(a ( ))

= a
l
(i
( )
)

= a
l
(w
1 )

=
j
(z)

.

This
implies
ha

a
( )

= z

and
he e o e

x=y
W
Le
us
suppose
ha
xE
[BKp
;Xh1
P
and
le y
E
F-1
(c
1
(x)
)
The e
exis s
z
such ha
c(z)
= y
because
d(y)
=
O
.
We
ha e
ha

c1
(k(z))

=
c
1
(x)

.
The e o e
he e
is

w
E
[Bn
p
;

K(n
n
;n)
1

such
ha

b1
(w)

=x
"
k
1
(z)

.

Le

w
1
= B

wah
g

.
Then

b
1
(w
1
)

_

(x
"
k
-1 (z))
k
.
g
EW
I
OllOws
om
he
s anda d

P
p ope ies
o
ib a ions
há
1

w
(x
"
k
(z)

k
)

lies

in
he cen e
o

[B7cP
;Xn1

P

.
The e o e
b1
(k
w
1
)

=x
"
k(z)
_1
.
We ha e
also ha

b
1
(
k w
1
)

-
k(b
(
k
w
1
) )

"
This
implies
ha
x
Eim
k
.
I
es
o
show
he
heo em
o
an
a bi a y
nilpo en ,
p-local
space X
.
We
use
once mo e
he
Pos niko
owe
o
X
and
h
e
same
diag am
as
be o e
.
Themap
i
is
an
isomo phism
because
QX
n-1
is
a
loop
space
.
We
assume, ha
Q
is
su jec i e
.
To
show ha
k
is
su jec i e
wemus
use he
ollowing
lemma
.
Lemma
2
.
Le M
be
a
ini ely
gene a ed

Z
(P)
-module
.
Le
us
suppose
ha
he
abelian
g oup
.
M
ac s
on
a
se
X
in
such
a
way
ha
iso-
opy
subg oups
a e

Z(P)-submodules
o

M
.
Wedeno e
'
his
ac ion
by
*
.
Le
us
suppose
u he
ha
a
ini e
g oup
G
ac s
on
M
and
on
X
,
he
ac ion
o
G
on
M
is
.
Z(P)-linea ,
he
o de
o
G

is

k
e
z*
p)

and

h
9
*xg

=

(h*x) g
I

x,x
1
E
X
G
and

w*x
= x
l
 hen

(
£

k
W
9
)
*
X =
xl
g EG
P oo
.
w*x
= x
1
and

x,x
1
EX
G

imply
ha

wg
*x
= x1
 o
each

9E
G
wg*
(w-wg)
*x))

=
wg*x

implies
ha

(w-w
9
)
*x
= x

o
each

g
E
G
The e o e

(k £
(w-w
g
))*x
= x
.
Weha e
ha
k
£
wg+
k
£
(w-w
g
)=
w-
gEG

gEG gEG
This
implies

(k

£

w
g
)
*x
=
x
l
.
I~
gEG
The
ac ion
o

[Bi
p
;K(n
n
;n)]

on

[B
p
;Xn
]

sa is ies
he
assump ions
o
Lemma
2
.
Wep o e
ha
k is
su jec i e
in
he
same
way
as
o
a
double'loop
space
.
Weha e
ha

c
1
(k
(z)
)

= c
1
(z)

.
The e-
o e
he e
is w
such
ha
wkk(z)
=
x
.
I
ollows
om
Lemma
2
ha
(

£

k
(w,h

) )
*k
(z)

=
x

.

(

£

k
(w,h

)
)

=

j
(wl)

implies

ha
9
EW

9

9EW

9
P

P
k(w
i*
z)
= x
.
The
spaces
Bn
and
Bi
Pha e
only
ini e
homology
g oups
he e o e
weha e
isomo phisms
96
[Bn
;X]

lim[Bu,X
n
1
and

.
[Bn
p ;X]

Z
lim[Bn
p ,X
n
]

.

(I

{X
n
}
nEN

'is
n

n
an
in e se
sys em
o
p-comple e
spaces
hen
he
unc o
lim[
;X
]
nn
is
ep esen able
by
Sulli an
i
.e
.

lim[
;Xn
]
_
[
;Z]

and
Z =
holim
X
n
.
n
In
ou
case
holim(X)

= X

and
np
[B n

(o

Bn
p
)
;

x

(o

X
n
)
]

_

[B n

(o

Bi
p
)
;X
p
(o (X
n)p)
]
because
Bi
and
Bn
p
ha e
ini e
homo opy
g oups
.)
Weha e
he
ollowing
commu a i e
diag am
[B
;X]
lim[Bi
;X
R
:
n]
n
W
p

p 1

W
[Bn
p
,
X]

3
lim[Bnp

n
;X]
p
n
p
is
an
isomo phism,
b
is an
epimo phism
( esp
.
isomo phism
i
X
is
a
loop space)
and
p
1
is a
monomo phism
.
This
implies
ha
a
is an
epimo phism
( esp
.
isomo phism
i
X
is
a
loop space)
.
This
inishes
he
p oo o
Theo em
2
.
I
we analize
he
p oo s
ca e ully
hen
i
appea s
ha
in
ac
weha e
p o ed
much
mo e
gene al
esul
.
Le
us
suppose
ha
a
ini e
g oup G
ac s
homo opically
on
a
space

X

,i
.e
.
he e
is a
homomo phism

G -
n
0(e
(X)
)

whe e

e(X)
is
he
space
o
all
homo opy
equi alences
o
X
.
Le
us
suppose
ha
1
G
1
=
k
,
X
is
p-local
and
k
E
Z(
p)
.
By
he
esul
o
Cooke
he e
is
a
space
X
1
wi h
a
ee
ac ion
o
G
and
a
homo opy
equi alence
i
:
X
-
"
X
1
which
is
homo opy
equi a ian
wi h
espec
o
he
homo opy
ac ion
o G
.