TSUKUBA
J.
$MA^{ }I^{ backslash }H$
.
Vol.
19
No.
2
(1995),
305–327
MOORE
SPACES
IN
PROPER
HOMOTOPY
By
R.
AYALA,
E.
DOM ’INGUEZ,
A.
M ’ARQUEZ
and
A.
QUINTERO
Abs ac .
Moo e
spaces
a e
de ined
in
p ope
homo opy
heo y.
Some
esul s
on
he
exis ence
and
uniqueness
o
hose
spaces
a e
p o en.
An
example
o
wo
non
p ope ly
equi alen
Moo e
spaces
is
gi en.
In oduc ion.
The
pu pose
o
his
pape
is
o
p o ide
he
co ec
s a emen s
and
de ails
o
he
esul s
announced
in
[4].
Namely,
we
p o e
he
exis ence
o
p ope
Moo e
spaces
o
ypes
o
ype
$(S;n)$
o
ce ain
objec s
$S$
in
he
abelian
ca ego y
o
owe s
o
g oups
( ow-JZ
$b,$ $ llco ne b$
)
and
$n geqq 2$
(Theo em
2.9).
Ne e heless
objec s
can
be
o
p ojec i e
dimension
2
in
( ow-db,
$ io a Ab$
),
and
his
ac
de e mines
an
obs uc ion
o
he
uniqueness
o
p ope
Moo e
spaces
(Theo em
3.2).
In
ac ,
an
example
o
wo
non
p ope ly
equi alen
Moo e
spaces
is
gi en
in
Appendix
A.
As
a
consequence
o
Theo em
3.2
wo
su icien
condi ions
o
he
uni-
queness
o
p ope
Moo e
spaces
a e
s a ed
(Co olla y
3.7
and
P oposi ion
3.9).
The
exis ence
o
Moo e
spaces
in
p ope
homo opy
was
al eady
announced
in
[4],
bu
in
ha
pape
he
obs uc ion
om
Theo em
3.2
was
no
conside ed
and
he
uniqueness
o
such
spaces
was
w ongly
asse ed.
Fo
owe s
o
p ojec i e
dimension
1,
p ope
Moo e
spaces
beha e
in
a
e y
simila
way
o
o dina y
Moo e
spaces.
In
pa icula ,
o
p ojec i e
dimension
1,
p ope
Moo e
spaces
de ine
p ope
homo opy
g oups,
and
a
coe icien
exac
sequence
which
gene alize
he
a ious
p ope
homo opy
g oups
known
in
he
li e a u e
and
hei
co esponding
Milno
exac
sequences
(Examples
2.15).
1.
P elimina ies
and
No a ions
CATEGORIES
OF
TOWERS.
Gi en
a
ca ego y
$c$
,
he
ca ego y
o
owe s
o
$C$
,
ow-C,
is
he
ca ego y
o
in e se
sequences
$s= {A_{1} le a ow A_{2} le a ow cdo s }$
in
$C$
whe e
a
( ow-C)-mo phism
$ :S igh a ow S^{ p ime}$
is
ep esen ed
by
a
sequence
o
C-mo phisms
1980
Ma hema ics
Subjec
Classi ica ion.
$54C10,55P99,55Q70$
.
Key
wo ds:
Moo e
space,
homology
decomposi ion,
p ope
homo opy,
owe
o
g oups.
Recei ed
Augus
25,
1993.
306
R.
AYALA,
E.
$D0M NGUEZ$
,
A.
MARQUEZ
and
A.
QUINTERO
$ _{k}$
:
$A_{n_{k}} igh a ow A_{k}^{ p ime},$ $ n_{1}<n_{2}< cdo s$
,
such
ha
gi en
$ >s$
he e
exis s
$j>n_{ },$ $n_{s}$
making
commu a i e
he
diag am
$A_{n} upa ow^{ } unde line{ _{ }}A_{ }^{ p ime,}$
$A_{j}$
$A downa ow_{n},$
$ unde line{ _{1}}A$
;
whe e
he
maps
wi hou
name
a e
bonding
maps.
We
a e
in e es ed
in
he
ull
subca ego y
o
$ o albox{ small REJECT}_{o }$
( ow-C)
whose
objec s
a e
a ows
$ :X igh a ow A$
whe e
:IE7
is
a
( ow-C)-objec
and
$A$
is
a
C-objec
ega ded
as
a
cons an
owe
whose
bonding
maps
a e
he
iden i y.
This
ca ego y
is
deno ed
( ow-C,
$C$
).
A
( ow-C,
$C$
)-mo phism
om
$ :x igh a ow A$
o
$g: e a igh a ow B$
can
be
ega ded
as
a
$C$
-mo phism
be ween
$A$
and
$B$
and
a
( ow-C)-mo phism
om
ec
o
$Q $
such
ha
bo h
mo phisms
a e
compa ible
ia
he
bonding
maps.
I
is
con enien
o
ep esen
( ow-C,
$C$
)
as
ollows.
Objec s
a e
owe s
$X=$
$ {X_{0} le a ow X_{1} le a ow cdo s }$
;
a
mo phism
consis s
o
a
map
$ :X igh a ow Y$
in
ow-C,
oge he
wi h
a
compa ible
map
$ _{0}$
:
$X_{0} igh a ow Y_{0}$
in
$C$
.
We
shall
specially
use
he
abo e
cons uc ions
o
$c= ma hcal{F}_{0}p,$ $ ma hcal{G}_{ },$ $ llco ne lb$
;
he
ca e-
go ies
o
opological
spaces,
g oups
and
abelian
g oups
espec i ely.
Since
$ ow- ma hcal{A}6$
and
( ow-X6,
a6)
a e
abelian
ca ego ies
(see
[1])
we
can
de-
ine
ke nels
and
images
and
s a e
exac
sequences
in
a
na u al
way.
In
pa i-
cula ,
we
can
use
p ojec i e
objec s
and
de ine
he
unc o
Ex .
See
[10]
o
de ails.
PROPER
CATEGORY.
A
p ope
map
(p-map)
is
a
con inuous
map
$ :X igh a ow Y$
such
ha
$ ^{-1}(K)$
is
compac
o
each
compac
subse
$K subse eqq Y$
.
P ope
homo opy
(p-homo opy),
p ope
homo opy
equi alence,
e c
$ cdo s$
can
be
de ined
in
he
na u al
way.
We
shall
deal
wi h
he
ca ego y
$ ma hcal{P}$
o
$T_{2}$
-locally
compac
$ sigma$
-compac
spaces
and
p-maps.
One
can
check
ha
$ ma hcal{P}$
is
a
co ib a ion
ca ego y
in
he
sense
o
H.
Baues
(see
[5]
and
[3])
whose
co ib a ions
a e
he
$p$
-maps
wi h
he
P ope
Homo opy
Ex ension
P ope y.
We
call
hese
$p$
-maps
p ope
co ib a ions
(p-co ib a-
ions).
I
can
be
shown
ha
any
p-co ib a ion
is
a
closed
embedding.
We
deno e
p-co ib a ions
by
a ows
angle
$ igh a ow$
.
Moo e
spaces
in
p ope
homo opy
307
Gl en
a
space
$X$
in
$ ma hcal{P}$
,
a
sys em
o
$ in y$
-neighbou hoods
o
$X$
is
he
objec
o
$ ow- ma hcal{G}_{op},$ $ epsilon(X)= {U_{1} le a ow U_{2} le a ow cdo s }$
whe e
$ o e line{X-U}_{j}=K_{j}$
is
compac ,
$K_{j} subse eqq K_{j+1}$
and
$X=$
$ cup {in K_{j} ; j geqq 1 }$
.
We
ecall
ha
a
CW-complex
$X$
is
said
o
be
s ongly
locally
ini e
i
$X$
can
be
co e ed
in
a
locally
ini e
way
by
ini e
subcomplexes.
In
ha
case,
i
is
known
ha
$X$
admi s
a
coun able
locally
ini e
co e
by
ini e
subcomplexes
and
so
he
$ in y$
-neighbou hoods
o
$X$
can
be
chosen
o
be
subcomplexes.
Fini e
dimensional
locally
ini e
CW-complexes
and
locally
ini e
simplicial
complexes
a e
s ongly
locally
ini e
(see
[11]).
I
$ :X igh a ow Y$
is
a
p-map
and
$ {U_{j} }$
and
$ {W_{j} }$
a e
sys ems
o
$ in y$
-neighbou hoods
o
$X$
and
$Y$
espec i ely,
o
each
$j$
he e
exis s
$k(j)$
such
ha
$ (U_{k(j)}) subse W_{J}$
and
he e o e
we
ge
a
mo phism
$ epsilon( ): epsilon(X) igh a ow epsilon(Y)$
(see
[10]
o
mo e
de ails).
Gi en
a
space
$X$
in
$ ma hcal{P}$
a
F euden hal
end
o
$X$
is
an
elemen
o
he
se
$ ma hcal{F}(X)= lim_{ le a ow} pi_{0}(U_{j})$
whe e
$ pi_{0}(-)$
deno es
he
se
o
connec ed
componen s.
Now
le
$ ma hcal{P}^{J}$
be
he
ca ego y
$ ma hcal{P}$
unde
$ J=[0, in y$
)
such
ha
o
e e y
$ ma hcal{P}^{J_{-}}$
objec
$J igh a ow iX,$
$ io a$
is
a
p-co ib a ion.
The
ca ego y
$ ma hcal{P}^{J}$
is
a
ca ego y
o
co ib a-
ions
whe e
he
no ion
o
p ope
wedge
$(V_{p})is$
de ined
in
a
na u al
way.
The
se
o
p-homo opy
classes
in
$ ma hcal{P}^{J}$
will
be
deno ed
by
$[$
–,
$-]_{p}^{J}$
.
(1.0.1)
The
ca ego y
o e
$J,$ $ ma hcal{P}_{J}$
,
is
again
a
co ib a ion
ca ego y,
and
i
allows
he
de ini ion
o
p ope
quo ien s.
Mo e
explici ly,
i
$ :X igh a ow J$
is
a
$ ma hcal{P}_{J^{-}}$
objec ,
and
$A succ X$
he
p ope
quo ien
$X/pA( )$
is
de ined
by
he
push-ou
$ ln ma hcal{P}$
$ $
$_{1}I_{1}|q$
$ ee|$
$i$
$J succ---------*X/pA( )$
No ice
ha
$ o e line{i}$
is
a
p-co ib a ion
whose
image
is
$q(A)$
.
In
pa icula ,
$q(A)$
is
homeomo phic
o
$J$
.
Fu he mo e,
his
shows
ha
he
p ope
quo ien
has
he
same
o dina y
homo opy
ype
as
he
usual
opological
quo ien .
(1.0.2)
I
can
be
shown
ha
any
space
in
$ ma hcal{P}$
admi s
an
on o
p-may
$h:X igh a ow J$
([10;
6.5.3]).
Ac ually,
$h$
can
be
chosen
o
be
a
e ac ion
o
a
gi en
p-co ib a-
ion
$i:J subse eqq X$
.
Indeed,
since
all
$p$
-maps
on
$J$
a e
p-homo opic
one
can
ind
a
p-homo opy
$H:J imes I igh a ow J$
connec lng
$h|J$
o
$id_{J}$
.
Then
by
using
he
p ope
H.
E.
P.
one
ge s
a
$p$
-homo opy
$H^{ p ime}$
:
$X imes I igh a ow J$
ex ending
$H$
,
and
$H_{i}^{ p ime}$
is
a
p- e ac-
ion
o
$i$
.
(1.0.3)
The
p ope
homo opy
ype
o
$X/pA( )$
does
no
depend
on
he
map
308
R.
AYALA,
E.
DOM NGUEZ,
A.
MARQUEZ
and
A.
QUINTERO
$ ;X igh a ow J$
.
Mo eo e ,
gi en
ano he
p-map
$ ^{ p ime}$
:
$X igh a ow J$
,
he e
exis s
a
p-homo opy
equi alence
$ xi:X/pA( ) igh a ow X/pA( ^{ p ime})$
such
ha
$q ci c xi=q^{ p ime}$
.
This
is
due
o
he
homo-
opy
in a iance
o
push-ou s
in
co ib a ion
ca ego ies
([5:
II.
1.
$2b$
)]),
since
all
p ope
maps
on
$J$
a e
p ope ly
homo opic.
When
i
is
clea
which
map
$ $
is
in ol ed
in
he
quo ien ,
we
shall
d op
he
map
$ $
om
he
no a ion.
(1.0.4)
Finally,
le
$ ma hcal{P}^{*}$
deno e
he
ca ego y
unde
and
o e
$J$
.
I
can
be
shown
ha
$ ma hcal{P}^{*}$
is
a
co ib a ion
ca ego y
whe e
we
can
de ine
p ope
wedges
and
p ope
quo ien s
as
well
as
p ope
cones
$(c_{p})$
,
and
p ope
suspensions
$( Sigma_{p})$
.
Fo
a
space
$J succ X igh a ow J$
in
$ ma hcal{P}^{*}$
,
hese
cons uc ions
do
no
depend
on
he
p-
e ac ion
$ $
(up
o
p-homo opy
equi alence
in
$ ma hcal{P}^{J}$
).
Mo eo e ,
he
se
$[ Sigma_{p}X, Y]_{p}^{J}$
is
endowed
o
a
na u al
g oup
s uc u e
o
any
space
$Y$
in
$ ma hcal{P}^{J}$
.
See
[3]
o
de ails.
A
s ongly
locally
ini e
CW-complex
$X$
will
be
conside ed
as
a
space
in
$ ma hcal{P}^{*}$
by
choosing
a
cellula
embedding
$i:J subse eqq X^{1}$
,
and
a
$p$
- e ac ion
o
$i$
.
See
(1.0.2)
abo e.
I
(X,
$ alpha$
)
is
a
space
in
$ ma hcal{P}^{J}$
he
$( ow- ma hcal{G}_{ }, 9_{ })$
-objec
$ Pi_{n}(X, alpha)= { pi_{n}(X, *_{0})- pi_{n}(U_{1}, *_{1})- pi_{n}(U_{2}, *_{2}) cdo s }$ $(n geqq 1)$
is
called
he
n- h
homo opy
owe
o
he
pai
(X,
$ alpha$
),
whe e
$ alpha( _{j})=*J$
wi h
$ alpha([ _{j}, in y)) subse eqq U_{ }$
and
he
bonding
maps
a e
induced
by
he
inclusions
and
he
base-
poin
change
isomo phisms.
(1.0.5)
$A$
space
$X$
is
said
o
be
p ope ly
k-connec ed
i
$ ma hcal{F}(X)= {* }$
and
$ Pi_{ }(X, alpha) cong 0$
in
$( ow- ma hcal{G}_{ }, ma hcal{G}_{i})(0 leqq leqq k)$
.
Simila ly
o
p ope
pai s
(X,
$A$
)
wi h
$ ma hcal{F}(X)= ma hcal{F}(A)= {* }$
.
I
is
wo h
poin ing
ou
ha
al hough
$ alpha$
and
$ alpha^{ p ime}$
ep esen
he
same
F euden hal
end,
he
owe s
$ Pi_{n}(X, alpha)$
and
$ Pi_{n}(X, alpha^{ p ime})$
need
no
o
be
isomo phic
(see
[23;
p.
13]).
Ne e heless,
i
$X$
is
p ope ly
l-connec ed
he e
is
no
dependence
on
he
ay
$ alpha$
.
In
ac ,
any
$p$
-homo opy
$H: alpha cong alpha^{ p ime}$
induces
a
p o-isomo phism
$H_{ $}$
;
$ Pi_{n}(X, alpha^{ p ime}) igh a ow Pi_{n}(X, alpha)$
.
In
addi ion,
i
$H: cong g$
is
a
p-
homo opy
he e
is
a
commu a i e
diag am
$ Pi_{x}(X unde line{ alpha) o e line{g^{*} _{*} sim}},’ p od_{ Pi_{n}}n(Y, ci c alpha)(Y,g ci c alpha) downa ow G_{l}$
whe e
$G=H ci c( alpha imes id)$
.
The
owe s
$ Pi_{n}(X, A, alpha)(n geqq 2)$
a e
also
de ined
o
pai s
(X,
$A$
)
in
$ ma hcal{P}^{J}$
.
In
Moo e
spaces
in
p ope
homo opy
309
addi ion,
i
$A=J$
we
ha e
he
iden i ica ion
$ Pi_{n}(X, J; alpha)= Pi_{n}(X, alpha)(n geqq 1)$
.
We
ecall
ha
E.
B own
in
[7]
gi es
a
unc o
$P: ow- ma hcal{G}_{s} igh a ow ma hcal{G}_{ }$
which
ca ies
he
owe
$ Pi_{n}(X, alpha)$
o
he
B own-G ossman
g oup
$ pi_{n}^{ in y}(X, alpha)$
.
In
a
simila
way
we
can
de ine
a
unc o
$P_{0}$
:
$( ow- ma hcal{G}_{ }, ma hcal{G}_{i}) igh a ow ma hcal{G}_{ }$
which
maps
$ Pi_{n}(X, alpha)$
o
he
global
B own-G ossman
g oup
$ pi_{n}(X, alpha)$
(see
[17]).
The
n- h
homology
owe
o
$X$
can
be
de ined
as
he
owe
$(n geqq 0)$
$H_{n}(X)= {H_{n}(X)-H_{n}(U_{I})-H_{n}(U_{2})- cdo s }$
whe e
he
bonding
maps
a e
induced
by
he
inclusions.
The
chain
complex
o
owe s
o
$X,$ $C_{*}(X)$
is
$ { pa ial:C_{n}(X) igh a ow C_{n-1}(X) }$
whe e
$C_{n}(X)= {C_{n}(X) le a ow C_{n}(U_{1})-C_{n}(U_{2})- cdo s }$
.
Now
a
p ope
cohomology
heo y,
$H^{n}$
,
wi h
coe icien s
in
a
( ow-db,
$ cup b$
)
$-$
objec
$S$
is
de ined
as
he
homology
o
he
complex
$... le a ow C^{n}(X) le a ow C^{n-1}(X) le a ow cdo s$
whe e
$C^{n}(X)=( ow_{c}- Ab, Ab)(C_{n}(X), S)$
(see
[14]
o
de ails).
Also
ela i e
e sions
o
hese
unc o s
o
pai s
(X,
$A$
)
in
$ ma hcal{P}$
a e
de ined.
No ice
ha
he
abo e
unc o s
(including
$ Pi_{n}$
)
a e
well
de ined
up
o
( ow-
$ ma hcal{G}_{i},$ $ ma hcal{G}_{ }$
)-isomo phisms.
Fundamen al
esul s
on
homo opy
g oups
like
he
Blake s-Massey
Theo em,
he
F euden hal
Theo em
o
he
Hu ewicz
Theo em
can
be
ansla ed
o
p ope
homo opy
by
using
he
ollowing
p oposi ion
1.1.
$p_{ROPOSlT10N}.-([2;1.1])$
Le
(X,
$A$
)
be
a
connec ed
s ongly
locally
ini e
CW-pai
wi h
only
one
F euden hal
end
and
assume
ha
(X,
$A$
)
is
p ope ly
k-connec ed.
Then
he e
exis s
a
s ongly
locally
ini e
CW-pai
(X’,
$A^{ p ime}$
)
such
ha
i)
$X$
( espec i ely
$A$
)
is
a
s ong
de o ma ion
p- e ac
subcomplex
o
$X^{ p ime}$
( espec i ely
$A^{ p ime}$
).
ii)
$(X^{ p ime})^{k} subse eqq A^{ p ime}$
.
In
pa icula
i
$A=J,$ $X$
has
he
same
homo opy
ype
as
a
CW-complex
$X^{ p ime}$
wi h
$(X^{ p ime})^{k}=J$
.
We
ecall
ha
o
any
pai
in
$ ma hcal{P}$
and
any
p ope
map
$ ho:X igh a ow J$
,
he e
is
a
na u al
homeomo phism
$J cong q(A)$
,
whe e
$q:X igh a ow X/pA( ho)$
is
he
quo ien
(see
(1.0.1)).
The e o e,
o
any
ay
$ alpha:J igh a ow X$
he
map
$q$
induces
mo phisms
o
owe s

310
R.
AYALA,
E.
DOM NGUEZ,
A.
MARQUEZ
and
A.
QUINTERO
$q_{*};$ $ Pi_{ }(X, A, alpha)- Pi_{ }(X/pA( ho), q^{0} alpha)$
1.2.
THEOREM.–Le
(X,
$A$
)
be
a
s ongly
locally
ini e
CW-pai
such
ha
(X,
$A$
)
is
p ope ly
n-connec ed
and
$A$
is
p ope ly
m-connec ed
$(n, m geqq 1)$
.
Then
he
mo phism
$q*de ined$
abo e
is
an
isomo phism
i
$2 leqq leqq m+n$
,
and
an
epimo -
phism
i
$ =m+n+1$
.
PROOF.
Fi s ly,
we
shall
p o e
he
heo em
when
$X^{n} subse eqq A$
and
$A^{m}=J$
.
Le
$i:J igh a ow A$
be
he
inclusion.
Acco ding
o
(1.0.2),
we
can
ind
a
p ope
e ac ion
$ :X igh a ow J$
o
$i$
.
Le
$ {U_{j}^{ p ime} }$
be
a
sys em
o
$ in y$
-neighbou hoods
o
$X$
con-
sis ing
o
subcomplexes.
Wi hou
loss
o
gene ali y
we
can
assume
ha
$ (U_{j}^{ p ime})$
$ subse eqq[ _{j}, in y)$
.
Le
$ U_{j}=U_{j}^{ p ime} cup[ _{ }, in y$
).
I
is
clea
ha
$ {U_{j} }$
is
a
new
sys em
o
$ in y-$
neighbou hoods
wi h
$ U_{j}^{1}=[ _{j}, in y$
),
$(U_{j}, U_{j} cap A)$
is
n-connec ed
and
$A_{j}=U_{j} cap A$
is
m-connec ed
o
any
$j geqq 0$
.
Then,
i
$X/pA$
is
he
p ope
quo ien
cons uc ed
wi h
he
e ac ion
$ $
,
i
is
easily
checked
ha
$ {U_{j}/pAj }$
is
a
sys em
o
$ in y-$
neighbou hoods
o
$X/pA$
,
whe e
$U_{ }/pAj$
is
cons uc ed
by
using
he
es ic ion
$ |U_{j}$
:
$ U_{j} igh a ow[ _{j}, in y$
).
Since
p ope
quo len s
has
he
same
o dina y
homo opy
ype
as
o dina y
quo ien s,
we
can
le elwise
apply
he
o dina y
Blake s-Massey
Theo em
[24;
6.22]
o
ge
isomo phisms
$q_{j* ddagge } pi_{ }(U_{j}, A_{j}) igh a ow pi_{ }(U_{j}/pA )$
i
$ leqq n+m$
and
epimo phisms
i
$ =m+n+1$
.
Now
he
esul
ollows
when
$ alpha=i$
and
$ ho= $
.
Mo eo e ,
we
can
use
he
na u ali y
o
he
base
ay
change
isomo -
phisms
and
he
homo opical
in a iance
o
p ope
quo ien s
(see
(1.0.3),
and
(1.0.5))
o
p o e
he
esul
o
(X,
$A$
)
as
abo e
and
a bi a y
$ alpha$
and
$ ho$
.
The
gene al
case
can
be
educed
o
he
p e ious
case
by
using
Theo em
1.1.
Using
Theo em
1.2
and
a
p oo
simila
o
he
o dina y
case
([24;
6.23])
we
ob ain
1.3.
THEOREM.–Le
$X$
be
a
p ope ly
n-connec ed
s ongly
locally
ini e
CW-
complex.
Then
he e
is
a
na u al
suspension
$( ow- ma hcal{G}_{ }, ma hcal{G}_{ })$
-mo phism
$ Sigma_{*}:$ $ Pi_{k}(X)- Pi_{k+1}( Sigma_{p}X)$
which
is
an
isomo phism
i
$k leqq 2n$
and
an
epimo phism
i
$k=2n+1$
.
1.4.
REMARK.–The
esul s
s a ing
ha
$ Sigma$
and
$q$
induce
isomo phisms
be-
ween
he
co esponding
homology
owe s
can
be
p o ed
in
a
s aigh o wa d
way
wi hou
using
P oposi ion
1.1.
Moo e
spaces
in
p ope
homo opy
311
P oposi ion
1.1
and
he
abo e
ema k
gi e
an
easy
p oo
o
he
ollowing
heo em.
1.5.
THEOREM.
$-$
([21;
II.4.2.7])
Le
$X$
be
a
p ope ly
n-connec ed
s ongly
locally
ini e
CW-complex.
Then
he
na u al
Hu ewicz
mo phism
$h; Pi_{k}(X) igh a ow H_{k}(X)$
is
an
isomo phism
i
$k=n+1$
and
an
epimo phism
i
$k=n+2$
.
Fu he mo e
Theo em
1.2
and
Rema k
1.4
p o ide
a
ela i e
e sion
o
Theo em
1.5
o
a
p ope
pai
$(X, A)$
o
s ongly
locally
ini e
CW-complexes
wi h
$A$
p ope ly
l-connec ed
and
$X$
p ope ly
n-connec ed.
Finally
Theo em
1.5,
he
B own-G ossman
unc o
$P_{0}$
in
(1.0.5),
and
[7;
$p$
,
$43J$
lead
o
1.6.
THEOREM.–Gi en
a
p-map
$ :X igh a ow Y$
whe e
$X$
and
$Y$
a e
p ope ly
1-
connec ed
ini e
dimensional,
locally
ini e
CW-complexes
such
ha
$ _{*};$ $H_{ }(X) igh a ow H_{ }( }^{7})$
is
an
isomo phism
o
each
$ $
,
hen
he
map
$ $
is
a
p-homo opy
equi alence.
2.
P ope
Moo e
Spaces.
We
s a
wi h
some
no ions
in
(
$ ow_{c}- Ab,$
(Ab).
2.1.
DEFINITION.–A
ee
owe
in
( ow-.A6,
A6)
is
a
owe
$F( ma hcal{L})= dagge F(L_{0})-F(L_{1})- cdo s }$
whe e
he
ollowing
ou
condi ions
hold;
i)
$ ma hcal{L}$
is
a
il a ion
$ ma hcal{L} equi L_{0} supse eqq L_{1} supse eqq cdo s$
wi h
$L_{0}$
a
coun able
se .
ii)
$ bigcap_{j=1}^{ in y}L_{j}= emp yse $
.
iii)
The
di e ences
$L_{k} backslash L_{k+1}$
a e
ini e.
i )
$F(L_{i})$
is
he
ee
g oup
gene a ed
by
$L_{i}$
and
he
bonding
mo phism
a e
in-
duced
by
he
inclusions.
Gi en
he
owe s
$F( ma hcal{L})$
and
$F( ma hcal{L}^{ p ime})$
,
i
can
easily
be
checked
ha
any
bijec-
ion
$L_{0} cong L_{0}^{ p ime}$
induces
an
isomo phism
$F( ma hcal{L}) cong F( ma hcal{L}^{ p ime})$
in
( ow-
A6,
$ ma hcal{A}b$
).
So
he
iso-
mo phism
class
o
$F( ma hcal{L})$
is
de e mined
by
he
ca dinali y
o
$L_{0}$
.
2.2.
REMARKS.
$-a$
)
F ee
owe s
a e
p ojec i e
objec s
$ ln$
( ow-Ab,
$ ma hcal{A}b$
)
(see
[14]).
b)
Gi en
a
s ongly
locally
ini e
CW-complex
$X$
,
and
a
sys em
o
$ in y$
-neigh-
bou hoods
$ {U_{j} }$
consis ing
o
subcomplexes,
he
owe
o
cellula
n-chains
o
$X$
,
$C_{n}(X)= {C_{n}(X)-C_{n}(U_{1})-C_{n}(U_{2})- cdo s }$
is
ob iously
a
ee
owe .
Also
he
owe
o
cellula
n-cycles
312
R.
AYALA,
E.
DOMiNGUEZ,
A.
MARQUEZ
and
A.
QUINTERO
$Z_{n}(X)= {Z_{n}(X)-Z_{n}(U_{1}) le a ow Z_{n}(U_{2}) le a ow cdo s }$
is
a
ee
owe .
Mo e
gene ally,
he
ke nel
o
any
mo phism
be ween
wo
ee
owe s
is
always
a
ee
owe
(see
[14;
5.1]).
2.3.
DEFINITION.–A
owe
$S$
is
said
o
be
geome ically
admissible
i
he e
exis s
an
exac
sequence
in
( ow-Ab,
$ ma hcal{A}b$
)
$j_{s}$ $j_{2}$ $j_{1}$
$0-F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$
When
$F( ma hcal{L}_{3})$
is
i ial
we
say
ha
$S$
has
geome ical
p ojec i e
dimension
(g.p.
$d.$
)
1.
0 he wise,
we
w i e
$g$
.
$p.d$
.
$S=2$
.
2.4.
REMARK.–Fo
any
s ongly
locally
ini e
CW-complex
$X$
he
n- h
homology
owe
o
$X$
is
geome ically
admissible
since
he
exac
sequence
o
owe s
$0-Z_{n+1}(X)-C_{n+1}(X)-Z_{n}(X)-H_{n}(X)-0$
is
a
ee
esolu ion.
Howe e ,
he
sho
exac
sequence
$0 igh a ow{ m Im} pa ial_{n+1} igh a ow Z_{n}(X) igh a ow H_{n}(X) igh a ow 0$
is
no
always
a
ee
esolu ion.
Indeed,
le
$X$
be
he
CW-complex
ob ained
om
he
cylinde
$ S^{n} imes[0, in y$
)
by
a aching
an
$(n+1)$
-cell
a
$S^{n} imes {j }$
by
a
map
$ :S^{n} igh a ow S^{n}$
o
deg ee
$2^{j}(j geqq 1)$
.
Then
one
can
check
ha
${ m Im} pa ial_{n+1}$
is
no
ee
since
$ pa ial_{n+1}$
has
no
igh
in e se.
2.5.
REMARK.–In
[9]
Dymo
in oduced
he
no ion
o
cop esen a ion
o
a
owe
$S$
.
Mo e
explici ly,
ollowing
[9]
we
say
ha
$s= {G_{0} le a ow G_{1} le a ow cdo s }$
admi s
a
cop esen a ion
i
he e
exis
a
le elwise
epimo phism
$ phi:F( ma hcal{L}) igh a ow S$
wi h
$F( ma hcal{L})$
a
ee
owe
and
subse s
$R_{ } subse eqq F(L_{ })$
wi h
$R_{j}-R_{J+1}$
ini e
and
$ cap in y R_{j}= emp yse $
,
such
$j Le a ow 1$
ha
$Ke phi_{j}$
is
he
subg oup
$ langle R_{j} angle$
gene a ed
by
$R_{j}$
.
In
gene al,
he
owe
$ langle R angle$
$ equi langle R_{0} angle le a ow langle R_{1} angle le a ow cdo s$
needs
no
o
be
p ojec i e.
I
is
easy
o
check
ha
a
owe
$S$
is
geome ically
admissible
i
and
only
i
admi s
a
cop esen a ion
(up
o
isomo phism).
In
ac ,
i
$ langle 9 angle igh a ow F( ma hcal{L}) igh a ow S igh a ow 0$
is
a
Dymo
cop esen a ion,
and
$F(R)$
deno es
he
ee
owe
consis ing
o
he
ee
g oups
$F(R_{i})$
,
he e
is
a
le elwise
epimo phism
$ F(R) igh a ow k langle R angle$
and
an
exac
sequence
$F(R)^{ k} igh a ow langle R angle igh a ow F( ma hcal{L})$
.
By
Rema k
2.2(b),
$Ke ( ci c k)$
is
a
ee
owe ,
and
so
$S$
is
geome ically
admissible.
Con e sely,
o
any
ee
esolu ion
$j_{3}$ $j_{2}$ $j_{1}$
$0 igh a ow F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$
Moo e
spaces
in
p ope
homo opy
313
le
$ { a phi_{i} : F(L_{n(i)}^{2}) igh a ow F(L_{ }^{1}) }$
be
a
le elwise
ep esen a i e
o
$j_{2}$
.
Then
$s^{J}=$
{Coke
$ a phi_{i}$
}
is
isomo phic
o
$S$
by
exac ness
and
admi s
he
ollowing
cop e-
sen a ion.
Le
$ phi:F( ma hcal{L}_{1}) igh a ow S^{ p ime}$
be
he
na u al
le elwise
quo ien
mo phism.
We
ake
$R_{i}= a phi_{i}(L_{n(i)}^{2}) subse eqq F(L_{i}^{1})$
.
Now,
i
is
clea
ha
$R_{i}-R_{i+1} subse eqq a phi_{i}(L_{n(i)}^{2}-L_{n i+1}^{2})$
is
ini e.
I
$ ilde{S}^{n}$
(
$ ilde{B}^{n}$
espec i ely),
is
he
space
ob ained
by
a aching
ini ely
many
copies
(possibly
no
copy)
o
$S^{n}$
(
$B^{n}$
espec i ely)
$(n geqq 2)$
a
each
$ m in N subse eqq[0, in y$
),
one
checks
ha
$ Pi_{n}( ilde{S}^{n})( Pi_{n}( ilde{B}^{n}, ilde{S}^{n-1})$
espec i ely)
can
be
iden i ied
in
a
na u al
way
wi h
some
$F( ma hcal{L})$
,
wi h
$L_{0} subse eqq N$
.
Mo eo e
he
ollowing
esul
holds:
2.6.
LEMMA.–Gi en
a
space
$X$
in
$ ma hcal{P}^{J}$
,
he e
is
a
na u al
bijec ion
$ ho:[ ilde{S}^{n}, X]_{p}^{J}$
$ cong( ow- A6, Ab)(F( ma hcal{L}) igh a ow Pi_{n}(X))(n geqq 2)$
,
gi en
by
$ ho([ ])= * cdo $
Mo eo e ,
i
$Z$
is
he
mapping
cone
$X bigcup_{ } ilde{B}^{n+1}$
in
he
co ib a ion
ca ego y
$ ma hcal{P}^{J}$
,
hen
$ *can$
be
also
ega ded
as
he
bounda y
ope a o
$d_{n+1}$
;
$ Pi_{n+1}(Z, X) igh a ow Pi_{n}(X)$
.
PROOF.–Take
$ a phi:F( ma hcal{L}) igh a ow Pi_{n}(X)$
.
A e
iden i ylng
$ Pi_{n}( ilde{S}^{n})$
wi h
$F( ma hcal{L})$
,
le
$ pi_{n}( ilde{S}^{n})- pi_{n}( ilde{S}_{k(1)}^{n})- pi_{n}( ilde{S}_{k(2)}^{n}- cdo s$
$ downa ow$ $ downa ow$
$ downa ow$
$ pi_{n}(X)- pi_{n}(U_{1})- pi_{n}(U_{2})- cdo s$
be
a
le elwise
ep esen a i e
o
$ a phi$
,
whe e
$ ilde{S}_{k( )}^{n}$
is
ob ained
by
dele ing
om
$ ilde{S}^{n}$
he
copies,
$S_{j}^{n}$
,
o
$S^{n}$
placed
a
he
poin s
$1 leqq in <k( ),$ $ k(1)<k(2)< cdo s$
.
We
de ine
$ |S_{j}^{n}$
as
a
ep esen a i e
o
$ a phi_{h(j)}[l_{ }]$
whe e
$l_{j}$
:
$S_{j}^{n} subse eqq ilde{S}_{k( )}^{n},$ $ k( ) leqq$
$j<k( +1)$
.
One
easily
checks
ha
$ _{*}= a phi$
.
This
p o es
ha
$ ho$
is
on o.
The
injec i i y
ollows
in
a
simila
way.
Finally,
we
ha e
he
diag am
$ Pi_{n+1}(Z, X) Pi_{n+1}( ilde{B}^{n+1} unde line{p_{ cong^{*}}} ilde{S}^{n})$
$ Pi_{n} downa ow_{(X)}$ $- Pi_{n}( ilde{S}^{n})=F( ma hcal{L}) downa ow cong$
whe e
$p: ilde{B}^{n+1} igh a ow Z$
is
he
canonical
p-map
and
$p_{*}$
is
an
isomo phism
by
Theo em
1.2
since
$ ilde{B}^{n+1}/p ilde{S}^{n} cong ilde{S}^{n+1} cong Z/pX$
.
Fo
he
sake
o
simplici y,
we
shall
use
he
single
no a ion
$ ilde{S}^{n}( ilde{B}^{n})$
o
all
he
“s ings”
o
sphe es
(balls)
desc ibed
abo e.
The
pa icula
objec s
$ ilde{S}^{n}( ilde{B}^{n})$
we
a e
using
in
he
u u e
will
be
clea
om
he
con ex .
Simila ly
o
$F( ma hcal{L})$
.
320
R.
AYALA,
E.
DOMiNGUEZ,
A.
MARQUEZ
and
A.
QUINTERO
is
he
p-map
de ined
by
he
cha ac e is ic
maps
o
he
$(n+1)$
-cells
o
$X$
.
We
easily
check
ha
$ _{n+1*}= a phi_{n+1}$
.
When
one
ies
o
go
u he
an
obs uc ion
appea s
as
ollows.
In
he
diag am
$ Pi_{n+2}(X^{n+2}, X^{n+1})$ $ Pi_{n+1}(X^{n+1})$
$ pa ial_{n+2}- Pi_{n+1}(X^{n+1}, X^{n})$
$ a phi_{n+2} downa ow$
(1)
$ downa ow _{n+1*}$ $ downa ow _{n+1*}= a phi_{n+1}$
$ Pi_{n+2}(Y^{n+2}, Y^{n+1})$
$ Pi_{n+1}(Y^{n+1})- Pi_{n+1}(Y^{n+1} pa ial_{n+2}^{ p ime}Y^{n})$
he
squa e
(1)
needs
no
be
commu a i e.
One
de ines
an
elemen
in
$( ow- Lambda b, Lambda b)$
$(F( ma hcal{L}_{s}); Pi_{n+1}(Y^{n+1}))$
by
he
di e ence
$ be a( _{n+1})= _{n+12} pa ial_{n+2}- pa ial_{n+2}^{ p ime} a phi_{n+2}$
Since
he
o he
squa e
is
commu a i e
we
ha e,
by
de ini ion
o
$Ke j_{*}$
in
( ow-
$ llco ne Ab,$ $ ma hcal{A}b$
)
ha
$ be a$
can
be
ega ded
as
a
mo phism
$ be a( _{n+1}):F( ma hcal{L}_{ he a}) igh a ow Ke j_{*}= Gamma_{n+1}Y$
.
Ob iously
$ be a$
is
a
cocycle
and
de ines
a
class
$c( a phi) in H^{n+2}(X, Gamma_{?l+1}Y)$
.
The
nex
lemma
shows
ha
$c( a phi)$
is
a
well
de ined
obs uc ion.
3.3.
LEMMA.
$-1$
)
$c( a phi)$
does
no
depend
on
he
mo phisms
$ a phi_{i}(i=n, n+1, n+2)$
.
2)
$c( a phi)$
is
an
obs uc ion
o
ealizing
$ a phi$
.
PROOF.
1)
Le
$ { a phi_{i}^{ p ime} }$
be
ano he
mo phism
such
ha
he
diag am
$(^{*})$
com-
mu es
and
le
$ _{n+1}^{ p ime}$
:
$X^{n+1} igh a ow Y^{n+1}$
be
a
$p$
-map
ealizing
$ a phi_{n+1}^{ p ime}$
.
I
is
a
well-known
ac
om
Homological
Algeb a
in
abelian
ca ego ies
ha
$ { a phi_{i} }$
and
$ { a phi_{i}^{ p ime} }$
a e
homo opic
chain
mo phisms.
Thus,
he e
exis
mo phisms
$ { alpha_{i} : C_{i}(X) igh a ow C_{i+1}(Y) }$
$(i=n, n+1, n+2)$
such
ha
he
ollowing
equali ies
hold
a)
$ a phi_{n}^{ p ime}- a phi_{n}=d_{n+1}^{ p ime} ci c alpha_{n}$
;
b)
$ a phi_{n+1}^{ p ime}- a phi_{n+1}= alpha_{n} ci c d_{n+1}+d_{n+2}^{ p ime} ci c alpha_{n+1}$
;
and
c)
$ a phi_{n+2}^{ p ime}- a phi_{n+2}=d_{n+3}^{ p ime} ci c alpha_{n+2}+ alpha_{n+1} ci c d_{n+2}$
.
By
c)
and
he
de ini ion
o
$ be a( _{n+1})$
we
ha e
(I)
$ be a( _{n+1}^{ p ime})- be a( _{n+1})=( _{n+1*}^{ p ime}- _{n+1^{*}}- pa ial_{n+2}^{ p ime} ci c alpha_{n+1} ci c j_{*}) ci c pa ial_{n+2}$
whe e
$j_{*}:$ $ Pi_{n+1}(Y^{n+1}) igh a ow C_{n+1}(Y)$
.
Now,
a)
p o ides
a
$p$
-homo opy
$H:X^{n} imes I$
$ igh a ow Y^{n+1}$
be ween
$ _{n}$
and
$ _{n}^{ p ime}= _{n+1}^{ p ime}|_{X^{n}}$
.
As
in
o dina y
homo opy
heo y,
$H$
yields
a
“di e ence”
mo phism
$ Del a=d( _{n+1}^{ p ime}, H, _{n+1});C_{n+1}(X)- Pi_{n+1}(Y^{n+1})$
(see
[15]).
Mo eo e ,
$H$
can
be
chosen
in
such
a
way
ha
$ in * ci c Del a= _{n+1*}^{ p ime}- _{n+15}$
$- alpha_{n} ci c d_{n+1}$
.
Take
$ be a= Del a- pa ial_{n+2}^{ p ime} ci c alpha_{n+1}$
.
By
b),
$j_{*}( be a)=0$
.
And
he
p ojec i eness
o
$C_{n+1}(X)$
allows
us
o
ega d
$ be a$
as
an
elemen
in
( ow-.A6,
$ cup b$
)
$(C_{n+1}(X); Gamma_{n+1}(Y))$
since
$Ke j_{*}= Gamma_{n+1}(Y)$
.

Moo e
spaces
in
p ope
homo opy
321
Finally,
one
can
eadily
check
om
he
de ini ions
ha
$ _{n+1}^{ p ime} cong p _{n+1}+$
$( be a+ pa ial_{n+2}^{ p ime} ci c alpha_{n+1})$
and
hen,
by
Lemma
3.1
he
igh
side
in
he
equali y
(I)
is
$ be a^{Q}j_{*}o pa ial_{n+2}= be a ci c d_{n+2}= del a be a$
.
This
p o es
$[ be a( _{n+1})]=[ be a( ex {{ i ’{n}}}_{+1})] in H^{n+2}(X; Gamma_{n+1}(Y))$
.
2)
I
$c( a phi)=0$
,
le
$w in( ow- Ab, llco ne Ab)(C_{n+1}(X), Gamma_{n+1}(Y))$
be
such
ha
$ be a( _{n+1})=$
$ del a w=w ci c d_{n+2}=w ci c j_{*} ci c pa ial_{n+2}$
.
Take
$ o e line{ }_{n+1}= _{n+1}+w$
.
By
Lemma
3.1,
$ be a( o e line{ }_{n+1})=$
$( _{n+1*}-w ci c j_{*}) pa ial_{n+2}- pa ial_{n+2}^{ p ime} ci c a phi_{n+2}=0$
,
and
$ o e line{ }_{n+1}$
ex ends
o
a
p-map
$ _{n+2}$
:
$X igh a ow Y$
wi h
$ _{n+2*}= a phi:H_{n}(X) igh a ow H_{n}(Y)$
.
3.4.
REMARKS.
$-a$
)
The
obs uc ion
$c( a phi)$
was
al eady
conside ed
by
J.
H.
C.
Whi head
in
o dina y
homo opy
(see
[25;
S 6])
and
i
can
be
de ined
wi hin
he
gene al
se lng
o
co ib a lon
ca ego ies
(see
[5;
VII.
1.13]).
b)
Fo
any
$n geqq 3$
he e
a e
wo
non
p ope ly
equi alen
Moo e
spaces
o
ype
$(S;n)$
.
The
examples
a e
gi en
in
Appendix
A.
As
an
immedia e
consequence
o
Theo em
3.2
we
ha e
3.5.
COROLLARY.–I
$S$
is
a
owe
$ w io a$
h
g.p.d.
$S=1$
hen
he e
exis s
(up
o
p-homo opy)
a
unique
p ope
Moo e
space
o
ype
$(S, n)(n geqq 2)$
.
Mo e
gene ally,
we
can
s a e
3.6.
COROLLARY.
–I
$S$
is
a
geome ically
admz
ssible
owe
wi h
$Ex ^{2}(S; Gamma_{n}(S))$
$=0$
,
hen
he e
exis s
a
unique
p ope
Moo e
space
o
ype
$(S, n)(n geqq 2)$
.
He e
$ Gamma_{n}(S)$
deno es
he
owe
ob ained
om
$S$
by
applying
le elwise
he
algeb aic
Whi ehead
$ Gamma_{n}$
- unc o
(see
$ lceil_{-}26$
;
Ch.
II]
o
[5;
IX.
4]).
I
is
known
ha
$ Gamma_{n}=- o imes Z_{2}$
when
$n geqq 3$
.
So,
(3.6)
yields
3.7.
COROLLARY.
–I
$S$
is
a
geome ically
admissible
owe
wi h
$Ex ^{2}(S;S o imes Z_{2})$
$=0$
hen
he e
exis s
a
unique
p ope
Moo e
space
o
ype
$(S, n)$
o
all
$n geqq 3$
.
PROOF
$0F(3.6)$
.
Le
$Y$
be
a
p ope
Moo e
space
o
ype
$(S, n)$
cons uc ed
as
in
he
p oo
o
Theo em
2.9.
Le
$ Y supse eqq U_{1} supse eqq cdo s supse eqq U_{n} cdo s$
be
a
sys em
o
$ in y-$
neighbou hoods
such
ha
each
$U_{j}$
is
a
subcomplex.
Mo eo e ,
each
$U_{j}$
is
$(n-1)$
-connec ed
by
cons uc ion.
Thus
by
[19;
VIII.
2.4]
o
$n geqq 3$
and
[26;
III.
14]
o
$n=2$
we
ha e
$ Gamma_{n+1}U_{j} cong Gamma_{n}(H_{n}(U_{j}))$
and
he e o e
$ Gamma_{n+1}Y cong Gamma_{n}(S)$
.
I
$X$
is
ano he
p ope
Moo e
space
o
ype
$(S;n)$
we
can
ealize
id:
$S igh a ow S$
by
a
$p$
-map
$ :X igh a ow Y$
by
Theo em
3.2
since
$H^{n+2}(X; Gamma_{n+1}Y)=Ex ^{2}(S; Gamma_{n+1}Y)=0$
.
By
Theo em
1.6
$ $
is
ac ually
a
p-homo opy
equi alence.
322
R.
AYALA,
E.
$DoMINGUEZ$
,
A.
M ’ARQUEZ
and
A.
QUINTERO
3.8.
REMARK.–The
owe
$s= {Z_{2} le a ow Z_{4}p_{1} le a ow Z_{8} le a ow p_{2} ldo s }$
whe e
$p_{i}(1)=1$
,
has
geo-
me ical
p ojec i e
dimension
2
since
$S$
is
he
n- h
homology
owe
o
he
CW-
complex
gi en
in
Rema k
2.4.
Ne e heless
$ Gamma_{n}S$
is
he
cons an
owe
$Z_{2}$
when
$n geqq 3$
,
and
$ Gamma_{2}S$
is
isomo phic
o
$S$
since
$ Gamma_{2}Z_{2n}=Z_{4n}$
acco ding
o
[26;
II.(B)].
Then,
one
can
check
as
in
Appendix
A
ha
$Ex ^{2}(S; Gamma_{n}S)=0$
.
So,
he e
is
a
unique
p ope
Moo e
space
o
ype
$(S;n)(n geqq 3)$
by
Co olla y
3.7.
The
same
esul
holds
o
$n=2$
.
Ano he
su icien
algeb aic
condi ion
on
$S$
o
he
uniqueness
o
p ope
Moo e
spaces
o
ype
$(S;n)n geqq 3)$
is
he
ollowing.
3.9.
$p_{ROPOSITION}.$
–Le
$S$
be
a
geome ically
admissible
owe
such
ha
$To ^{1}(S;Z_{2})=0$
.
Then
he e
is
a
unique
Moo e
space
o
ype
$(S;n)(n geqq 3),$
uni-
que
up
o
p-homo opy.
Be o e
s a ing
he
p oo
o
P oposi ion
3.9
we
shall
ix
no a ion
and
p o e
a
lemma
whose
p oo
$1S$
simila
o
he
p oo
o
[14;
Lemma
2].
Le
$( ow- Z_{2}, Z_{2})$
deno e
he
abelian
ca ego y
de ined
in
he
same
way
as
(
$ ow- ma hcal{A}b$
,
A6)
by
using
$Z_{2}$
- ec o
spaces
ins ead
o
abelian
g oups.
Gi en
a
ee
owe
$F( ma hcal{L})$
,
le
$Z_{2}(L_{i})$
deno e
$F(L_{i}) o imes Z_{2}$
.
Then
3.10.
LEMMA.–Any
owe
$ {V_{0} le a ow V_{1} le a ow V_{2} cdo s }$
wi h
$V_{i} subse eqq Z_{2}(L_{i})$
and
wi h
bond-
ing
mo phisms
he
co esponding
es ic ions
is
p ojec i e
in
$( ow- Z_{2}, Z_{2})$
.
PROOF.
We
may
ind
a
basis
$T_{i}$
o
$V_{i}/V_{i+1}$
such
ha
any
elemen
o
$T_{i}$
is
ep esen ed
by
a
linea
combina ion
o
elemen s
in
$L_{i}-L_{i+1}$
.
Le
$B_{i}$
be
he
union
$ cup {T_{j} ; j geqq I }$
.
I
is
easy
o
check
ha
$B_{i}$
is
a
basis
o
$V_{i}$
.
Since
$ B_{0} supse eqq B_{1} cdo s$
and
$ cap B_{i}= emp yse $
i
is
s aigh o wa dly
shown
ha
$ {V_{0} le a ow V_{1} le a ow V_{2} cdo s }$
is
p ojec i e.
PROOF
OF
PROPOSITION
3.9.
Le
$d_{2}$ $d_{2}$ $d_{0}$
$0-C_{2}-C_{1}-C_{0}-S-0$
be
a
ee
esolu ion
o
$S$
.
Since
$S o imes Z_{2}$
is
a
owe
o
g oups
o
o de
2
we
ha e
an
isomo phism
( ow-
A6,
$Ab$
)
$(C_{i} ; S o imes Z_{2}) cong( ow- Z_{2}, Z_{2})(C_{i} o imes Z_{2} ; S o imes Z_{2})$
The e o e,
$Ex ^{2}(S;S o imes Z_{2}) cong Ex ^{1}({ m Im} d_{1} ; S o imes Z_{2}) cong Coke ((d_{2} o imes 1)^{*})$
Moo e
spaces
in
p ope
homo opy
323
whe e
$*$
s ands
o
he
dual
mo phism.
Now
om
he
exac
sequence
$0-C_{2}-C_{1}-{ m Im} d_{1}-0$
we
ge
he
exac
sequence
$0-C_{2} o imes Z_{2}$
$d_{2} o imes 1 igh a ow C_{1} o imes Z_{2}{ m Im} d_{1} o imes Z_{2} unde line{d_{1} o imes 1} igh a ow 0$
since
he
owe
$Ke (d_{2} o imes 1)$
is
isomo phic
o
he
owe
$ {To ({ m Im} d_{1}^{i} ; Z_{2}) }$
which
is
i ial
because
each
componen
${ m Im}[d_{1}^{i} : C_{1}^{k(i)} igh a ow C_{0}^{i}]$
o
he
owe
${ m Im} d_{1}$
is
a
ee
abelian
g oup.
Thus,
Coke
$((d_{2} o imes 1)^{*}) cong Ex _{Z_{2}}^{1}({ m Im} d_{1} o imes Z_{2} ; S o imes Z_{2})$
whe e
he
igh
side
is
he
$Ex 1$
unc o
in
he
ca ego y
$( ow- Z_{2}, Z_{2})$
.
On
he
o he
hand
we
ha e
he
commu a i e
diag am
$0-C_{2} o imes Z_{2}$ $C_{1} o imes Z_{2}{ m Im} d_{1} o imes Z_{2}$
$d_{2} o imes 1 unde line{d_{1} o imes 1}-0$
$ Ve $ $ Ve $
$0-c_{2} o imes z_{2^{-}}^{d_{2} o imes 1}$ $C_{1} o imes Z_{2} igh a ow^{d_{1} o imes 1}{ m Im}(d_{1} o imes 1)-0$
whe e
he
uppe
ow
is
exac
as
i
was
p o en
abo e.
And
he
lowe
ow
is
also
exac
since
$To ^{1}(S;Z_{2})=Ke (d_{1} o imes 1)/{ m Im}(d_{2} o imes 1)=0$
by
hypo hesis.
Thus
${ m Im} d_{1} o imes Z_{2} cong{ m Im}(d_{1} o imes 1)$
and
hence
$Ex _{Z_{2}}^{1}({ m Im} d_{1} o imes Z_{2} ; S o imes Z_{2}) cong Ex _{Z_{2}}^{1}({ m Im}(d_{1} o imes 1);S o imes Z_{2})$
.
Now
he
o me
e m
anishes
because
${ m Im}(d_{1} o imes 1)$
is
p ojec i e
by
Lemma
3.10.
This
yields
$Ex ^{2}(S;S o imes Z_{2})=0$
and
he
uniqueness
ollows
om
Co olla y
3.7.
FINAL
REMARK.
The
ca ego y
o
ees
o
abelian
g oups
(see
[12])
seems
o
be
he
igh
algeb aic
amewo k
o
a
gene aliza ion
o
he
esul s
o
his
pape
o
spaces
wi h
many
F euden hal
ends.
Appendix
A.
Two
non
p ope ly
equi alen
p ope
Mo e
spaces
o
ype
$(S;n),$ $n geqq 3$
.
Le
$S$
be
he
owe
$ { bigoplus_{1}^{ in y}Z_{2} oplus Z_{2} igh a ow^{k_{1} oplus 1} bigoplus_{2}^{ in y}Z_{2}Z_{2} }$
$k_{2} oplus 1$
$ ldo s$
in
( ow-di,
$ ma hcal{A}b$
),
wi h
$k_{j}$
s anding
o
he
na u al
inclusion
mo phism.
A
ee
esolu ion
o
$S$
is
324
R.
AYALA,
E.
$DoM NGUEZ$
,
A.
M ’ARQUEZ
and
A.
QUINTERO
$ pa ial_{2}$ $ pa ial_{2}$
$0-F( ma hcal{L}_{3})-F( ma hcal{L}_{2})-F( ma hcal{L}_{1})-S-0$
whe e
$ ma hcal{L}_{3}$
is
$ L_{1}^{3} supse eqq L_{2}^{3} supse eqq cdo s$
wi h
$L_{j}^{3}= { alpha_{i} ; i geqq] }$
.
$ ma hcal{L}_{2}$
is
$ L_{1}^{2} supse eqq L_{2}^{2} supse eqq cdo s$
wi h
$L_{j}^{2}=$
$ { ho_{i}, mu_{i}, sigma_{i} ; i geqq j }$
and
$ ma hcal{L}_{1}$
is
$ L_{1}^{1} supse eqq L_{2}^{1} supse eqq cdo s$
wi h
$L_{j}^{I}= { epsilon_{i}, gamma_{i} ; i geqq j }$
.
And
he
mo -
phisms
a e
gi en
by
$ pa ial_{2}( alpha_{i})= mu_{i+1}-2 sigma_{i}- mu_{i}$
;
$ pa ial_{1}( ho_{i})=2 epsilon_{i},$ $ he a_{1}( mu_{i})=2 gamma_{i}$
and
$ pa ial_{1}( sigma_{i})=$
$ gamma_{i+1}- gamma_{i}$
.
Clea ly,
$S=S_{1} oplus S_{2}$
,
whe e
$S_{1}$
is
he
owe
$ { bigoplus_{1}^{ in y}Z_{2} bigoplus_{2}^{ in y}Z_{2} } unde line{k_{1}} unde line{k_{2}} ldo s$
and
$S_{2}$
is
he
cons an
owe
$ {Z_{2}=Z_{2}= cdo s }$
.
A.
1.
LEMMA.
$-Ex ^{2}(S;S) neq 0,$ $Ex ^{2}(S_{1} ; S_{1})=Ex ^{2}(S_{2} ; S_{2})=0$
.
PROOF.
By
he
na u ali y
o
$Ex $
’
and
$ oplus$
we
ha e
$Ex ^{2}(S;S)= oplus$
$ {Ex ^{2}(S_{i} ; S_{j});i, j leqq 2 }$
.
On
he
o he
hand,
i
is
easy
o
check
ha
$g$
.
$p$
.
$d$
.
$S_{1}=1$
,
and
so
$Ex ^{2}(S_{1} ; S_{j})$
$=0$
.
Wi h
he
abo e
no a ions,
$S_{2}$
admi s
he
ee
esolu ion
$ pa ial_{2}$ $ pa ial_{1}$
$0 igh a ow F( { alpha_{i} })-F( { mu_{i}, sigma_{i} })-F( { gamma_{i} })-S_{2}-0$
and
by
he
s anda d
Hom-Ex
exac
sequence
we
ge
$Ex ^{2}(S_{2} ; S_{2})=Ex ^{1}({ m Im} pa ial_{1} ; S_{2})=0$
.
Indeed,
o
any
$ a phi in( ow- Ab, d6)(F( { alpha_{i} }), S_{2})$
we
may
de ine
$ o e line{ a phi} in$
(
$ ow- Ab,$
Ab)
(
$F( { mu_{i}, sigma_{i} }, S)$
by
$ o e line{ a phi}( sigma_{i})=0, o e line{ a phi}( mu_{1})=0$
and
$ o e line{ a phi}( mu_{J})= Sigma { a phi( alpha_{i});i leqq j-1 }$
.
Then
$ o e line{ a phi} ci c pa ial_{2}$
$= a phi$
.
Finally
$Ex ^{2}(S_{2}, S_{1})=Ex ^{1}({ m Im} pa ial_{1} ; S_{1}) neq 0$
since
$ xi:F( { alpha_{i} }) igh a ow S_{1}$
gi en
by
$ xi( alpha_{i})=$
$ epsilon_{i} in y 1$
de ines
a
non- i ial
elemen .
0 he wise,
$ xi= au ci c pa ial_{2}$
o
some
$ epsilon;F( { mu_{i}, sigma_{i} })$
$ igh a ow S_{1}$
and
$ au$
yields
he
equali ies
$ epsilon_{i} o imes 1= au( mu_{i+1})- au( mu_{i})(i geqq 1)$
.
As
$ au$
is
a
p o-
mo phism
one
can
induc i ely
p o e
ha
$ au( mu_{i}) in bigoplus_{k geq 1}Z_{2}$
and
he
sequence
$ { epsilon_{i} o imes 1 }$
would
ep esen
he
i ial
elemen
$ ln lim^{1}S_{I}$
and
i
is
a
well-known
ac
ha
i
does
no .
Lemma
A.1
and
Co olla y
3.7
yield
ha
$R$
(
$S_{1}$
;
n)
and
$R$
(
$S_{2}$
;
n)
a e
uniquely
de e mined
up
o
p-homo opy
$(n geqq 3)$
.
Ac ually
hese
ypes
a e
ep esen ed
by
$ ilde{W}$
and
$ W imes[0, in y$
),
whe e
$W^{ app ox}$
is
ob ained
by
a aching
one
copy
o
$W$
a
each
na u al
coo dina e
o
$[0, in y$
)
and
$W=S^{n} bigcup_{2}e^{n+1}$
is
he
n-sphe e
wi h
an
$(n+1)-$
cell
a ached
by
a
map
o
deg ee
2.
Thus,
$X=R(S_{1} ; n)_{p}R$
(
$S_{2}$
;
n)
is
a
ep e-
sen a i e
o
$R(S;n)$
by
Co olla y
2.10.
A.2.
LEMMA.
The
na u al
map
$[X; X]_{p}^{J} igh a ow( ow- AA, Lambda b)(S;S)$
is
on o.
Moo e
spaces
in
p ope
homo opy
325
PROOF.
By
P oposi ion
2.14
$[R(S_{1} ; n);X]_{p}^{J} igh a ow[ ci c$
is
on o.
On
he
o he
hand,
( ow-db,
$Ab$
)
$(S_{2} ; S_{1})= lim S_{1}=0 le a ow$
and
$[R(S_{2} ; n);X]_{p}^{J} igh a ow$
$[R(S_{2} ; n);R(S_{2} ; n)]_{p}^{J} cong[W;W]^{J} igh a ow_{ ze a}A6(Z_{2} ; Z_{2}) cong( ow- Ab, ma hcal{A}b)(S_{2} ; S_{2})$
.
whe e
he
i s
bijec ion
is
gi en
by
he
Edwa ds-Has ings
embedding
Theo em
([10;
6.27]).
Finally
he
na u al
bijec ion
$[X; X]_{p}^{J} cong[R(S_{1} ; n);X]_{p}^{J} imes$
$[R(S_{2} ; n);X]_{p}^{J}$
comple es
he
p oo .
Now
we
choose
a
non- i ial
elemen
$ alpha in H^{n+2}(X; Gamma_{n+1}X) cong Ex ^{2}(S;S) neq 0$
.
By
Rema k
2.7
we
may
assume
$X= ilde{B}^{n+2} bigcup_{h_{0}}X^{n+1}$
.
Now,
he
commu a i e
dia-
g am
$C_{n+2}(X)= Pi_{n+2}(X, X^{n+1} Pi_{n+2}( ilde{B}^{n+1} ec{h} cong_{*}S^{n+1})$
$ downa ow d_{n+2}$ $ cong downa ow d_{n+2}$
$ Pi_{n+1}(X^{n+1}) Pi_{n+1}(S^{n+1}) unde line{h_{0*}}$
allows
us
o
iden i y
he
bounda y
ope a o
$d_{n+2}$
wi h
he
mo phism
$h_{0*}(h$
is
he
cha ac e is ic
map
$h: ilde{B}^{n+2} igh a ow X$
).
The
isomo phism
$d_{n+2} ci c h_{*}^{-1}$
also
gi es
he
iden i ica ion
(I)
( ow-Ab,
$ cup b$
)
$(C_{n+2}(X); Pi_{n+1}(X^{n+1}))$
$ cong( ow- Ab, ma hcal{A}b)( Pi_{n+1}( ilde{S}^{n+1}); Pi_{n+1}(X^{n+1})) cong[S^{n+1} ; X^{n+1}]_{p}^{J}$
whe e
he
second
isomo phism
is
gi en
by
Lemma
2.6.
Thus,
i
$ alpha=[a],$ $a$
can
be
ega ded
as
a
p-map
$g;S^{n+1} igh a ow X^{n} subse eqq X^{n+I}$
.
Le
$ o e line{h}_{0}$
be
a
ep esen a i e
o
$[h_{0}]+[g] in[S^{n+1} ; X]_{p}^{J}$
and
le
$Y$
be
he
p ope
cone
o
$ o e line{h}_{0}$
.
Since
${ m Im} g subse eqq X^{n}$
,
he
complexes
owe s
o
cellula
chains
o
$X$
and
$Y$
a e
he
same.
Bu
A.3.
LEMMA.–The
obs uc ion
$c(id) in H^{n+2}(X; Gamma_{n+1}Y)$
gi en
in
Theo em
3.2
o
id:
$H_{n}(X)=S igh a ow S=H_{n}(Y)$
is
non- i ial.
PROOF.
Smce
$c(id)$
does
no
depend
on
he
mo phisms
$ a phi_{i}$
:
$C_{i}(X) igh a ow C_{i}(Y)$
inducing
id:
$S igh a ow S$
(see
Lemma
$3.3(1)$
),
one
can
conside
$ a phi_{i}=id$
o
each
$i=n$
,
$n+1,$ $n+2$
.
So,
$c(id)$
is
ep esen ed
by
$ be a(id)=d_{n+2}-d_{n+2}^{ p ime}$
,
whe e
$d_{n+2}$
is
gi en
in
he
abo e
diag am
o
$X$
.
Simlla ly
$d_{n+2}^{ p ime}$
o
$Y$
.
Bea ing
in
mlnd
he
iden i ica ion
(I)
$ be a(id)$
is
ega ded
as
$h_{*}-h_{*}-g_{*}=-g_{*}$
.
Then
$ be a(id)$
is
ac ually
$-a$
and
$c(id)=- alpha neq 0$
.
Finally
we
ge ,

326
R.
AYALA,
E.
DOMiNGUEZ,
A.
M ’ARQUEZ
and
A.
QUINTERO
A.4.
$p_{ROPOSITION}.-X$
and
$Y$
a e
no
p-homo opically
equi alen .
PROOF.
I
$h:X igh a ow Y$
is
a
p-homo opy
equi alence,
le
$h^{ p ime}$
be
a
$p$
-homo opic
in e se
o
$h$
.
The
mo phism
$h_{*}^{ p ime}$
:
$H_{n}(Y)=S igh a ow H_{n}(X)=S$
can
be
ealized
by
a
h-
map
$ :X igh a ow X$
acco ding
o
Lemma
A.2.
Then
$h ci c :X igh a ow Y$
is
a
$p$
-map
wi h
$(h ci c )_{*}=id:H_{n}(X)=s igh a ow s=H_{n}(1^{ nea ow})$
,
and
his
canno
happen
by
Lemma
A.3.
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1160
41080-SEVILLA
(SPAIN)
E.
Dom ’inguez
Dp o
$Ingenie ^{ p ime}1a$
El ’ec ica
$e$
In o m ’a ica
Facul ad
de
Ciencias
Ciudad
Uni e si a ia
50009-ZARAGOZA
(SPAIN)