Pub
.
Ma
.
UAB
Vol
.
28 Nó
1
Maig
1984
Equi a ian
maps
be weenG-spaces
induce
ibe
p ese ing
maps
be ween
he
associa ed
Bo elspaces
.
We
will
show
ha
no all
ibe
p ese ing
maps
be ween
Bo el
spaces
a e
induced
ha
way,
no
e en
all
ibe
homo opyclasses
o
such
maps
.
Howe e
he e
is a
one- o-one
co espondencebe weenhomo opyclasses
o
G.
-maps
(i .e
.
maps
equi a ian
up
o
homo opy
in-á way,
see
sec ion
1
o
de ini ions)
be ween
G-spaces
and
ibe
homo opyclasses
o
maps
be ween
Bo él
spaces
.
This
one- o-one
co espondence
is
ob ained
by
a
unc o
equi alence
be ween
he
espec i e
ca ego ies(Theo em
1
and
2
in
sec ion
4)
.
As
a
esul
equi a ian
homo opy
heo y
(in
amodi ied
sense)
is
equi alen
o
he
heo y
o
homo opy
ib a ions
.
To
p o e hese
heo ems
we ha e o includeH-spaces
in o
ou
discussion
:
In
ac ,
he
unc o
equi alence
men ioned
abo e
is
an
ex ension
o
he
equi alence
be ween
he
ca ego ies
o
H-spaces
and
classi ying
spaces
p esen ed
in
[2)
.
The e o e
we need
he
no ion
o
a
Bo el
space
o
H-spaces
.
EQUIVARIANT
MAPS
UP TO HOMOTOPY
AND
BOREL
SPACES
Ma in
Fuchs
The
Bo elspacewe
use,
is
associa ed
wi h
he
modi ied
Dold-Lasho
cons uc ion
in
[3)
.
In
sec ion
se enwe
p esen
a
numbe o
examples
o
G-spaces
wi h
di e ing
ix
poin
se s,
such ha
hese
di e ences
canno
be
de ec ed
by s udying
he
cohomology
o hei Bo elspaces,
no
by
s udying
he
Bo el
space
i sel
.
The
g oups
in
mos examples
a e
M
p
o
S
l
,
bu he
G-spaces
a eno all
o
ini e
dimension
.
Thus
we
illus a e
he
limi s
o
heo ems
like
he
local-
iza ion
heo em
by Hsiang
([5],
p
.
47)
.
All
he
examples
a ise
om
he
ac ha
i

h
=

(h
n
)

n
=
o,l,
.
.
.

is

a
G
.-map
be ween
he
G-spacesX
1
and
X
2
and
h
is
an
o dina y
homo opy
equi alen e,
hen
he
ibe
map
induced
be ween
he
Bo elspaces
is
a
ibe
homo opy
equi alen e
.
1
.
De ini ions
1
.1
.
The
H-spacesH
we
a e
using
a e
supposed o
be
s ic ly
associa i e
and
o ha e
a
s ic uni
elemen
e
.
Fu he mo e
we assume
H
has
a homo opy
in e se
(such
ha

H-=-->
H
x
H
lx ~
H
xH
u
>
H

is
homo opic
o

id
H
)
.
1
.2
.
We
say
ha
a
opological
space
X
is
a
G-space,
i
an
H-space
H
ac s
on
X
om
he
le
con inuously
and
in
a
s ic ly
associa i e
manne
.
We assume
ha
ex
=
x
o all
x
E
X
.
1
.3
.
As
usual,
an
H
m
-map
h
om
H
1
o
H
2
(o
leng h
)
is
asequence
o
con inuous
maps
h
n
:

(H
1x

I
)
n
x
H1

_#
H
2
(n
=

0
,1
,
2
, .
.
.
)

such

ha
h
n
(g0 ,
i 1
..
.1
n
.g
n
)
o

n

>

O,

g
0
.
. .
.,g
n

E

H
l
,

and

i
,
.
. .
,
n

E

I
=
[O, ]
c
7R
.

I
=
O,
hemap
h0
is
a
homomo phism
in
he
usualsense
.
11-4
.
I
H
1
ac s
on
X
1
and
H2
ac s
on
X
2
om
he
le ,
and
i
h
is
an
H
-map
om
H
1
o
H
2
o
a
leng h

,

henwe de ine
aG
C7
-map


om

X
1
 o
X
2
o leng h
associa ed
wi h
h
o
be
a
seguenceo
maps
n
:
(H 1 x I
)
n
x
X
1-0
X
2
(n
=
0,1,2

.
.)
such ha
o
n >
O
n
(g
0
, l

.
.,g
n-
l,
n
,x)
h
n-l
(g
0
, l

.
.,gi-lgi,
. .
., n,gn)
i
=
h
i-l
(g
0
, i
,.
.
.1g
i-1
)hn-i(gi,
.
.
. .
gn
)

=
i
=O
n-1(g0, l,
.
.
.,gi-1gi,
.
.
.,gn-l, n,x)
i
=
hi-l(g0,
.
.
.,gi-1
) n-i(gi,
.
.
.,gn-l, n,x)

i
=
0
Composi ion
o
H
-maps
and
G
-maps
is
de ined
as in
[3]
.
1
.5
.
I
is
a G
-map om
X
1
o
X
2
associa ed
o
he
H
-map
h
om
H
1
o
H
2
,
hen
is
called
a
m
G-
-homo opy
equi alence
i
he e
exis s
an
H
-
-map
k
om
H
2
o
H
1
and
a G.
-map
g
om
X
2
o
X
1
associa ed
wi h
k
such
ha

g
o

and

og
a e
G
-homo opiC
o
and
1

. .
2
o
he
H
m
-homo opies
be ween

k
oh

espec i ely

h
o
k
and

id
H

espec i ely

id
H
) .
1

2
We
a e
going
o
use he
heo em
om
[4]
:
Theo em
.
I
-
H
1
ac son
X
1
and
H2
ac s
on
X
2
and
i

h
:H
1
-o
H
2
is
an
H
m
-map
such ha

h
0
is
an
o dina y
homo opy
equi alence,
and
i
:X
1
-4
X
2 is
a
G
.
-map
associa ed
wi h
h
such
ha
0
is
an
o dina y
homo opy
equi alence,
hen

h

is
an
H
m
-homo opy
equi alence
and
is
a G
-homo opy
equi alence
associa ed
o
h
.
1 .6
.
H-spaces
and
H
-maps
o m
he
ca ego y
W
and
G-spaces
and
G_-maps
o m
he
ca ego yJ
, .

The
associa ed
homo opy
ca ego ies
a é
deno ed
by
1,!
and
.1
.
2
.
Co
:
.
.s uc ion
o
he
Bo el
Space
In
his
sec ion
we ely
hea ily
on [3J,
whe e
many
addi ional
.de ails
can
be
ound
.
2 .1
.
Le
(p, )
be
an
H-p incipal
ib a ion
E
xH

E
P l~
~P
E

B
espec i ely
(associa ed
as
desc ibed
in
[3]
and
le
X
be
a
G-space
wi h
espec
o

H

wi hac ion

s
:
H
xX
-+
X
.

Assume
ha

pX
:
EX
-+
B
is a
ib a ion
wi h
ibe
X
associa ed
o
p
:
E
-
i
B

in
he
ollowing
sense
:
1)
The wo
ib a ions
a e
ibe
homo opy
i ial
wi h
espec
o
he
same
nume able
co e ingZ
o
B
and
e e y
U
E
21
is
con ac ible
in

B
.

2)
The e
is
,
a
map

X
:
E
XX
-+
EX

such ha
o
each
U
F
.
n!
he
diag am
is
commu a i e
((a,(3,aX,PX)
maps)
.

In
addi ion
we
wan
o be
commu a i e
.
UxHxX--
.
1xs~U,~X
1

l
(U)
xX

---~
P
X
p

(U)
x
E
x
H
x
X
E
xX
X
->

EX
a e
he
ob ious
coo dína e
2 .2
.
Fo
he
gene al
s ep
o
he
Bo el
space
cons uc ion
we
look
a
he
H-p incipal
ib a ion
(p, )
as
desc ibed
in
[3],
p
.
329-331
.
The
base
space
B
o
henew
ib a ion
is
he
mapping
coneo
p
:E
-i
B
wi h
he
coo dína e opology
.
We
conside
he
co e ing
o
B
consis ing
o

B1
=
(y
J,
1 '3]

and

B
2=
(y1
l
Le
pl
:
E
l
-o
B
1
 espec i ely

plX
:
E
l
X
-o
B
1
be
he
ib a ions
inducedby

(yl )
=
p(y),

he
map
collapsing
B
1
o
he
ange
space
B
o
he
mapping
cone
B
.
p1X
is
associa ed
o pl
i
we de ine
1X
:
E
l
x
X
-o

E
1
X

by
u he mo e
le
E
2
= B
2
xH

and

E
2
X=B
2
XX
.
De ine

2X
(Y
.L
,h,x)
=
(y
.i
,hx)
.

Ob iously
hese
ib a ions
a e
associa ed
.
We ecall om
[3),
p
.
330, ha
he
map
F
:p
-1
(B
1
(1
B 2
)

-o
p-1
(B
1
()

B
2
)

de ined
by
is
a s ic ly
equi a ian
ibe
homo opy
equi alence
.
We
de ine
he
associa ed
map
F
X
:
p2X
(B
1
(1
B
2
)
p
-1
(B1
n
B2)
by
Fx
is
a
map
o e
B 1
(1
B
2
and
a homo opy
equi alence
on
each ibe
( his
ollows om
diag am
(1)
and he
ac
ha
H
has
a homo opy
in e se)
and
hence
is
a
ibe
homo opy
equi alence
acco ding
o
Theo em
6
.3
in
[1]
.
2
.3
.
As
in
[3),
p
.
330
we
now
o m
he
mappingcylinde
o
F
and
o
FX
and
cons uc
he
H-p incipal
ib a ion
p
:É
-o
B
and
simila ly
he
associa ed
ib a ion
84
l(Y
1
.Yl ,x)
=
(Y
1 ,
x
(Yl
,x))
Y(Y
l
.
,n)
=
(Y
1
,YIIi
F
X
(y
.~
,x)
=
(y 1
,
X
(Y,x)
)
pX
:
EX
-+
BX
.

Wi h
he
help
o

Xl

and

X2

we
in
he
ob ious
manne
.
No
X
:EXX
cons uc
p oblem
a ises
since
he
diag am
(y
i
,h,x)

2X
-
>(y
-,
,hx)
(y
1
,yh,x)

-
1
(y
1
,
(yh,x)
)
lX

X
commu es
as
a
consequence
o
diag am
(2)
.
So
i
is
easy
o
see
ha
E
and
EX
a e
associa ed
.
2
.4
.
To
cons uc
he
Bo el
space
o
X
we
s a
ou
wi
h

pO
:
EO
-"
B
O
,

whe e

EO =
H

and

BO
=

[*)
=
poin ,
and
wi h
poX
:E
O
X
-+
B
O
,
whe e
EOX
=X
.
F om
pn
and
pnX
we
cons uc
pn+1
and
pn+l,X by
le ing
E
.+l
= E
n'
B
n+l
-
Bn
and
E
n+1
X=En
X
.
Ob iously
p
.
333
in
[3]
we
use
elescopes o inally
ge he
uni e sal
H-p incipal
ib a ion
p
H
:EH
4
BH
and he
associa ed
ib a ion
pX
:EX
-+
BH
.
We
call EX
he
Bo elspace
o
X
and
pX
he
Bo el
ib a ion
.
o
X
.
No ice
ha
i ial
ib a ion
wi h
ibe
h ough
hemap
he
di ec
limi o
he
maps
because
we used
he
elescope
cons uc ion
.
con inui y
o
H
in
[3],
p
.
333)
.
-i
2
3<
<
3
pn+1
-Pn
and
pn+1,X-
pnX a e
associa ed
.
As
on
pX
is
a
nume able,
locally
ibe
homo opy
X
associa ed
wi h
pH
X
is
essen ially
n,X
,
and
i
is
con inuous
(Compa e
he
X
-EHXX
-i
3
.
InducedM
aps
Be ween
Bo el-Spaces
3 .1
.
Be o e
we
can
discuss
G-Spaces,
we ha e
o
know
mo eabou H-Spaces
.
So
le
h
:H
1
-+
H
2
be an
H
e
-map
Be ween
he
H-Spaces

H
1
and

H
2
.

We
de ine
a
G
m
-map
E0
h
:E
0
H
1
-+
E0
H
2
as

E0h = h
.

(No e
ha
all he
Spaces
EH
ha e
a
igh ac ion,
so
he
no ion
o
G
-map
has
o
n

m
be
modi ied
acco dingly)
.

Alsowe
le

B
0h
:B
0
H
1
-+
B0H
2
be
he
i ial
map
.
Assume
ha
Eh
has
been
ex ended
o
a
G
-map
O

°'
Enh
:E
nH
1
-+
EnH
2
associa ed
wi h

h

and

B0h

has
been
ex ended
o
Bh
such ha
n
(We
will
call
a G
-map
wi h
his
p ope y
ibe
p ese ing)
.
On
E
nl
H
l
we de ine
and
p
n2
0
E
n
hk
(Y . l.gl
..
. .
.
k
.gk)
= Bnh
°
pnl(Y)
.
.
Fi s
we
ex end
B
nh
om
Bn
H
1
o
BnH
1
by
de ining
E
n1
h0
(Y
1
.Y
0
)
=
(E
nh0
(Y)
1
.
Enh0
(Y
O
))
En1h
k
(Y
1 ,y
0
,
i
,
91
..
. .
. k
.
gk)
o
k =
1,2,
.
.
.
Bn
h(y
~
.
)
=
(E
n
h0
(Y)
1
)
=
(E
nh0
(Y)
1 ,
E
nhk
(Y
0
. 1.91
..
.
.,
k
.g
k
)
Recall

( om

[3],

p
.

330)

ha

En2H1
=

(Bn2
H
1
XH1)

U
(B
ni

n B
n2
XI
X
H1)

and
de ine
Eñ
2
hk(y
l
,T,g
o
.
l


k,gk)
((E
n
h0
(y)
l ,
hk
(g
0
' l


k
.g
k
))
(E
n
h0
(y)i
. ,
2T,h
k
(g
0
. l


k
.gk)
when
O
S
T
S
2
and
3
< <
3
(E
n
h0
(y)
x ,
Enhk+l(y,2T
-
l,g0
. l
.
.. .
. k
.gk))
when
Z
S
T
S1

and
3
<
T
<
3
'
(when
T
=1
we
use
ha
Enhk+l(y,l,g0, 1,
. .
.)
E
n
hk
(yg
0
, i
.
.
.
.)
.
Hence
Eñ2hk
and
E
nl
h
oge he
induce
a G_-map

E
nh

om

En
H
1
 o

E
nH2

which
sa is ies
all
he
condi ions
men ioned
be o e
and
hence
we
ge
E
n+1
h
'
E
n+1
H
1
-4
E
n+1
H2

oge he
wi h

B
n+l
h
.

In
he
ob ious
manne
we
ob iain
he
G
0
-map

Eh
:EH
1
-
i
EH
2
associa ed
wi h
h
.
Because
o
ou
de ini ion
o Eñ
2
hk
on
he
mapping
cylinde
pa
o
EnH,
.
we only
ge E(h
oh')
is
G
-homo opic
o
Eh
oEh'

and
simila ly
B(h
oh')
y
W
Bh
o
Bh'
.

In
ac
he
G-homo opymen ioned
is
ibe
p ese ing
.
We
ge
he
implies ha
LpH
oK
is a
ibe
homo opy
equi alence
and
hence
SO
is
a homo opy
equi alence
.
Lemma
2
.
S
O
can
be ex ended
o an
H
-map
.
m
P oo
:

Le
K
¡E
0
H = K
IH
= K0
.

Thenwe
ha e
o
ind
maps

S
1
,
S2
,.
.
.

which
make

S
O
=
Lp
H
o
KO
:
H
-+
nBH
in o
an
H
-map
.

Assume
we
al eady
cons uc ed
m
S
i =
Lp
H
o
K
i (i
=
O,l,
. .
.,n)
.

Then

Sn+l

and
hence
K
n+
1

is
de
inedon

~H
(n
+
1)

h ough
he
maps

S i

and
K
i
espec i ely
(i
=
0,
.
.
.,n)
.
and
Associa ed
wi h
K
i
a e
he
maps
De ine
k
i

:
H
(i)

X
7R
+

-4

EH
T4(
;1
.
.
io+
y
]
.
wi h

ki
(g0, l,
.
. .
.
i,g
i~0)

=
*

and
k
i
(g0
"
i,
. .
., ,,g,,,)
=
g
0
...
g
i
o
Z
i(g0, l,
. .
., i'gi)
.
These
maps
de ine
kn+l
and
n+1

espec i ely
on

-3H(n
+
1)
.

Since

]R
+
is
con ac ible
we
can
ex end

n+l

o
all
o

H(n
+
1)
.
Then
we
can
ex end

k
n+l

o
all
o

H(n
+
1)

such
ha
kn+l(g0' l
"
..
' n+l'gn+l'0)
_
and
kn+1(g0' i,
.
. .
. n+1,gn+1' n+l(
..
.)) =
g0
...
gn+l'
Since
EH
is
.con ac ible
.
K
n+l

(kn+l' n+l)

and
S
n+l
= L
PH
oK
n+
1

Fo
u hé
de ailscompa e
[2J,
p
.
214-215
.
(No e
he
addi ion
o
pa hs
on
p
.
213
shouldbe e e sed
.)
5
.2
.
P oQosi ion
.
S
is
a
na u al
ans o ma ion
be ween
1
y
and
QB
.
P oo
:
In
he
diag am
S
L(EH
;EH,*)
LP
H
h
LEh
H'
K
L(EH'
;EH',*)
L
PH'
íé(BH,*)

nBh
)
C(BH'
.*)
he
lowe
po ion
commu es
o
all he
maps
o
LEh
.
To
see
ha
he
uppe
po ioncommu es
up
o
an
H
m
-
homo opy,
onehas
o
look
again
a
he
associa edmaps
in o
EFI'
.
Since
EH'
is
con ac ible,
all
ex ensions
necessa y
o
cons uc
he
H
m
-homo opy
be ween

LEh
o
K
and
Koh
can
be
ca ied
ou
.
Fu he
de ails
in
[2]
.
(In
[2)
he
G
-map
Eh
was
no
discussed
.
Ins ead
he
no ion
o
a
" egula "
H-homomo phism
had
o
be used
.
Now
EH
p o ides
he
homo opy
be ween
o mula
2
and
2a
on
p
.
217
in
2
,
ansla ed
om
igh
o
le
ac ions
.)
5
.
3
.
Wi h
S
ou
o
he
way
we de ine
o
any
G-space
X
:
We
al eady
know
ha

T
0
(y)
_
(*,y)

is a
homo opy
equi alence
.

We de ine

Tn
:
(H
xI)
n
x
X
-+
WE

as
wi h

O
S
n
S
n-1
(g
o
l.
.,
'
n-1,gn-1
)

and
O
S
6
S
n-l
- n
.

Recall

X
:
EH
x
X
-+
EX
.

We
ha e
WB
.
Tn(g0
; i,
.
.
.
. n
.x)
=

(pKn-1
(g
o
,
.
. .
.
n-1
'g
n-1
)(
n+
a),
-1(go,
. .
. .
n-1'gn-1
)( n),x))
Tn(go, 1,
.
. .
.
gn-l, n,x)
T0
:
X
-+
WE

as

T0 =
T
I
X
6
.
P oo
o
Theo em
2
I(
S
n-1(g0
.
V
. . . .
g
n
_
1
)
.x)

n = O
* .9
09
1

.
.
.

g
n-l
,
x)

n
=

The
"G
.
-homo opy"
be ween

LEh
oK

and

K
oh

implies
ha
T
is
a
na u al
ans o ma ionbe ween
1
2(
and
66-1
.
Le
.T
*
be
he
ca ego y
o
based
opological
spaces
X,
which
ha e
a
nume ableco e ing
2I
such
ha
e e y
U
E
2I
is
con ac able
in X,
and
based
con inuous
maps
.
Le
T
*
be
he
associa ed
homo opy
ca ego y
.
Rema k
.
I
is
easy
o
see
ha
o
e e y
H
in
Al
he
classi ying
space
BH
is in
T
*
.
.
In
p epa a ion
o he
p oo o
Theo em
2
we
lis
h ee
uni e sal
ib a ions
wi h
ibe
n(X,*)
o
X E
a
*
.
a)
Applica ion
o
he
modi ied
Dold-Lasho
cons uc ion
o
he
i ial
ib a ion
(2(X,*)
-+
leads
o
p
C2X
:
ESZX
-i
B
2X
b)
I
is
well-known
ha
P
I,
:
L(X
;X,*)
-0
X
also
classi ies
nume able
n(X,*)- ib a ions
.
c)
I
we
apply
he
modi ied
Dold-Lasho
cons uc ion
o
pL
o
b),
we
ge
again
a
uni e sal
ib a ion
P
EL
:
ELX
-4
BLX
All
h ee
cons uc ions
induce unc o s
om
7*
o
6
.2
.
The
inclusion
o
C(X,*)
as
dis inguished
ibe
o
pL
L(X
;X,*)
4
X
can
be
in e p e ed
as
a
p incipal
map
o
p incipal
ib a ions
and
hence
i induces
he
ibe
map
T
*,
.
E(QX)
p
OX
B(CIX)

B(LX)
Le
g
be
a
homo opy
in e se
o
T
.
which
is
a
p incipal
ibe
homo opy
equi alence
;
( , )
is
an
inclusion,
hence
P
OX
is
p incipal
ibe homo opy
equi alen
o
he
pullpack
o
p
LX
.
Fo
uni e sal
ib a ions
his
implies
is
a
homo opy
equi alence
.
As
a
esul ,
( , )
ep esen s
a
unc o
equi alence
be ween
he
unc o s
om
T
*
o
91
*
induced
by
a)
and
c)
.
6
.3
.
The
inclusion
L(X
;X,*)
k
-j
ELX
kj
BLX
is
a
ibe homo opy
equi alence
by
he
same
easoning
as
desc ibed
in
6
.2
.
So
(k,k)
ep esen s
a
unc o
equi alence
be ween
he
unc o s
a ising
om
b)
and
c)
.
6
.4
.
Now
conside a
ib a ion
p
:E
-4
X
om
he
ca ego y
7
*
.

The
associa ed
Hu ewicz- ib a ion
p
:E-+X
admi s
a
map
0
:
L(X
;X,*)
;c
WE

E
de ined
h ough
he
addi ion
o
pa hs,
whichmakes
E
a
look
alike
o
a
Bo el
space
associa ed
o WE
.
Assigning
o
p
he
Hu ewicz
ib a ion
p
induces
a unc o H

on
;

which
is
ob iously
equi alen
o
id
~ .

.

We
a e
now
going
o
show

BW
-
.H
.

Conside
he
,,
*

_-
diag am
o
Bo el
spaces
:
K
is
i duced
by applying
he
Bo el
space
cons uc ion
o
p
(an
ob ious
modi ica ion)
and
G
is
induced,by
L(X
;X,*)
xWE

k

ELX
xWE

g
x
l
~E,-?X
xWE
p
g,
he
homo opy
in e se
o
om
6
.2
.
(K,k)
and
(G,g)
ep esen
unc o
equi alences
associa ed
o
he
equi alences
(k,k)
and
(g,g)
discussed
in
6
.2
and
6
.3
.
Since
he
igh
sideo
he
diag am
ep esen s
BW
and
he
le side
ep esen s
H
,

he
p oo
is
comple e
.
7
.
Two
Applica ions
K ~
G
E

--j
EL(WE)
--~
E(WE)
g
7
.1
.

Le

G =

IR1
and

X
=
IR
2
.

Cons
ide
he
wo
IR
1
-spaces

X
1
and

X
2
de ined
by
he
wo
ac ions
p
l
:
IR
l
x
IR
2

-~

IR
2
,

p
1
( , e
lcp
)

=

e
l
(g+ )
~2
:
IR
l
x
IR
2
-+
IR2
,

P2
( , e
lqp
)

=
el
(9+
(1- )
)

The ix
poin
se
o

pl

is
jus
he
o igin
o

IR
2
and
he ix
poin
se
o
p2
is
he
o igin
and he
uni
ci cle
.
Ob iously
we
couldde ineac ionswi h
mo e
complica ed
ix
poin
se s
.
The
cons an
map
om
one
o
hesespaces
o
he
o igin
o
he
o he
is
an
equi a ian
map
which
is
also
an
o dina y
homo opy
equi alen e
.
I
induces
(acco ding
o
sec ion
ou )
ahomo opy
equi alen e
be ween
he
Bo el
spaces
o
he
wo
spaces
.
7
.2
.
a)

Le
P
be an
acyclic
ini e
polyhed on
wi h
non i ial
undamen al
g oup
.

Then
he
suspension

E
P
is a
con ac ible
2Z2
-space
wi h
ix
poin
se

P,

and
he
join

P *
S
1
is
a
cons ac ible

S1
o
Z?
p
-space
(p
y
2)
wi h
ix
poin
se
P
in
he
ob iousmanne
(no ice

P
*S
1
-
E
2
p)
.
b)
Le
P
be
any
ini e
polyhed on
.
The
ob ious
2Z2
-ac ion
on

E
P

can
be
ex ended
o

E
2
P

e c
.

so
ha

lim
En
P

is a
con ac ible
2Z2
-space
wi h
ix
n~
poin
se
P
.
sameby
ei e a ing
he
join
wi h

S
1
.
G-space
wi h
nonemp y
ix
poin
se
F,
e
.g
.
le
Y
be
one
o
he
spaces
men ioned
abo e
.
The
one
poin
union
Id
o
X
and
-
Y
o medby
iden i ying
wo
Fo

G =
2Z
P

(p
T
2)

and

G = S
1
we
can
do
he
7
.3
.
Le
G
be ei he
2Z

o
S1
andle
X
be
a
P
G-space
wi h
ix
poin s
.
Le
Y
be
a con ac ible
ix
poin s
is
a
new
G-space
in
he
ob ious
manne
and
he
inclusion
o
X
in o
I^1
is
an
equi a ian
map
and
also
an
o dina yhomo opy
equi alence
.
By
he
heo em
in
[4]
he
inclusion
ep esen s
an
isomo phism
in
.1
and
induces
a
ibe
homo opy
equi alence
be ween
BX
and
BW by sec ion
4
.
Hence
he
cohomology
o
hese
Bo el
spaces
ca ies
no
in o ma ion
abou
F
.
7
.4
.

Assume

G

is
ei he

2Z
kk
o

(S1
)
k
and

X1
,X2
a e
G-spaces
which
sa is y
he
assump ions
o
Bo el's
heo em
as
desc ibed
in
P oposi ion
1
o
Chap e
IV
in
[5J,
i
.e
.,
le
X
l
,X
2
be
pa acompac
G-spaces
wi h
ini e
cohomology
dimension
.
Le
:X
1
-#
X2
be
an
equi a ian
map
which
is
also
an
o dina yhomo opy
equi alence
.

Again

E
:EX
1i
EX
2

is
a
ibe
homo opy
equi alence
be ween
Bo elspaces
.
E
induces
isomo -
phisms
be ween
H
G
(X
2
)
and
HG
(X1
)
as
H
(BG)
modules
.
Hence
P oposi ion
1
on
p
.45
in
[5]
ells
us,
ha
IF
1
:F
.
1
=#
F
2
induces
an
isomo phism
o
he
cohomology
ings

H
*
(F
2
)
(9
k
RG
and

H
*
(F
1
)
®
kRG
o
he ix
poin
se s
F
1
and
F
2
.
T
.
Pe ie
in [7]
and
elsewhe e,
Ch
.
N
.
Lee
and
A
.
Wasse man
in
[6]
Na e
cons uc ed
exampleso such
maps
which
do
no
ha e
equi a ian
homo opy
in e ses
.
Hence
he
ibe
homo opy
in e se
o
E

is
no
induced
by
an
equi a ian
map
om
X2
o
X
1
.
This
answe s
he
opening
s a emen
o
he
in oduc ion
o
his
pape
.
Re e ences
Dold,
A
.,
Pa i ions
o
uni y
in
he
heo y
o
ib a ions,
Ann
.
o Ma h
.
78,
223-255
(1963)
.
2
.

Fuchs,
M.,Ve allgemeine e
Homo opie-Homomo phismen
und
klassi izie ende
Ráume,
Ma h
.
Ann
.
161,
197-230
(1965)
.
3
.
Fuchs,
M
.,
A
modi ied
Dold-Lasho
cons uc ion
ha
does
classi y
H-p incipal
ib a ions,
Ma h
.
Ann
.
192,
328-340
(1971)
.
4
.
Fuchs,
M
.,
Homo opy
equi alences
in
equi a ian
opology,P oc
.
Ame
.
Ma h
.
Soc
.
58,
347-352
(1976)
.
5
.
Hsiang,
W
.Y
.,
Cohomology
heo yo
opological
ans o ma ion
g oups,
Sp inge -Ve lag
1975
.
6
.
Lee,Ch
.
N
.
and
Wasse man,
A
.G
.,
On
he
g oups
JO(G),
Memoi s
Ame
.
Ma h
.
Soc
.
159
(1975)
.
7
.
Pe ie,
T
.,
Smoo h
S1
ac ions
and
bilinea
o ms,
Bull
.
Ame
.
Ma h
.
Soc
.
79,
1056-1059
(1973)
.
Rebu
e1'
15
de~se embAe
del
19&3
Depa men
o
Ma hema ics
MichiganS a e
Uni e si y
Eas
Lansing,
MI
48824
USA