Publicacions
Ma emá iques,
Vol
36
(1992),
715-724
.
A
bs ac
POWER-CANCELLATION
OF
CW-COMPLEXES
WITH
FEW
CELLS
IRENE
LLERENA
Dedica ed
o
Pe e
Menal,
in
memo iam
In his
pape ,
we
use he
ac
ha
he
ings
o
in ege
ma ices
ha e
he
powe -subs i u ion
p ope y
in
o de
o
ob ain
a
powe -
cancella ion
p ope y
o
homo opy
ypes
o
CW-complexes
wi h
one
cell in
dimensions
0 and 4n and a
ini e
numbe
o
cells
in
dimension
2n
.
In oduc ion
Cancella ion
in
homo opy
is
closely
ela ed
o
he
genus
.
Recall
ha
he
genes
o a
homo opy
ype
X
is
he
se o
all
homo opy
ypes
Y
such
ha
he
p-localiza ions
o
X
and
Y
a e
homo opy
equi alen
o
all
p imes
p
:
X
p
-
Y
p
[9]
.
I
X
and
Y
a e
in
he
same
genus,
we
shall
w i e
X
-Y
.
All he
known
examples
o
non-cancella ion
occu
o
homo opy
ypes
in
he
same
genus
.
A
ele an
esul
in
his
line
is
he
ollowing
.
Theo em
1
(Wilke son
[12])
.
Le
X,
Y,
W
be
nilpo en
co-H-
complexes
wi h
ini ely
gene a ed
homology
.
Then
i)
X
V
Y-
X
V
W

implies
Y-W,

and
k

k
ii)
X
-
V
Y

implies
X
-
Y
.
In
[10]
Mislin
cha ac e ized
he
genes
o
nilpo en
co-H-spaces
wi h
ini ely
gene a ed
homology
in
he
ollowing
way
X
-Y

i
and
only
i

X
V
V
Sn
;
-Y
V
V
Smj
,
whe e
he
dimensions
o
he
sphe es
a e
de e mined
by
he
homo opy
g oups
o
X
and
Y
.
A
simila
esul
had been
ob ained
by
Molna
[111
71
6

1
.
LLERENA
o
CW-complexes
wi h
wo
cells,
Sn
U
en'
.,
wi h
a aching
map
o
ini e
o de
in
7 ,,_
1
(S')
.
See
also
[7]
.
In
[1]
Boko
de elops
ma icial
me hods
in
o de
o
deal
wi h
spaces
S2n
U
el,
such
ha
he
a aching
map
is
o
in ini e
o de
.
This
lead
him
o
p o e
he
ollowing
.
Theo em
2
(Boko
[1])
.
Le
xi
=
s2n
U
i
and
only
i
i)
h=h'
H H
ii)
V
xi
-
V
Y
i
Theo em
3
.
Le
Yi,
=
S2n
U
9i
e
4n
'
H H
V
x
i
=
V
Y
,
H
H
VXi
-
VYi
k
Suppose
ha
i
,
gj
ep esen
elemen s o
in ini e
o de
in
7 4
n
_
1
(VS
2
n)
i
and
only
i
i
<
h
and
j
<
h'
.

Then
h+l
h+l
iii)

The e
exis s
a
pe mu a ion
T
o
{I,
. . . ,
h} such
ha
X
i
--
Y,(¡)
o alli<h
.
In
[8]
we
ied
o
ex end
Boko 's
esúl
o
complexes wi h
one
cell
in
dimensions
0
and
4n and k
cells
in
dimension
2n
.
Fo
n
>_
2,
he
esul
u ned
ou
no
o
be
ue
.
Howe e ,
i
we
es ic
ou sel es
o
he
p-local case,
he
heo em
holds
.
The
p oo
uses
a
echnical
lemma
ha
is
also
used
in
he
p oo
o
a
cancella ion
p ope y
o
he
ing
M
k
(7L
)
o ma ices o e
he
p-localiza ion
7L
o
he
ing
7L
([4,
Th
.2])
.
Fo ou
pu pose,
we
s a e
he
ollowing
e sion
o
he
main
heo em
in
[8]
.
k

k
x
i
=
V
S2n
U
i
e
4n

1
i
-
V
S2n
U9i
e4n

i
=
1)
. .
.,
H
.
Suppose
ha
i,
g7
ep esen
elemen s o
in ini e
o de
in
7F4n_1(Vk
S2n)
i
and
only
i
i
<
h
and
j
<
h'
.

Then
i
and
only
i
i)
h=h'
h h
ii)
VXi
_
VY,
H H
VX
i
-VY
i
.
h+l
h+l
In his
pape
we
use
a
p ope y
o
he
ings
M
k
(ZL),
he
(le )
powe -
subs i u ion p ope y,
o
p o e
he
ollowing
heo ems
.
Theo em
I
.
Le
POWER-CANCELLATION

717
k

k
X=VS-U e-,
Y=VSnU9e-
.
k
Assume
ha
,g
a e
suspension
elemen s o
ini e
o de
in
7,,_1(VSn)
.
Then
i
X
-
Y,
he e
is
a
posi i e
in ege
such
ha

VX-VY
.
Obse e
ha
he
con e se
holds
by
Theo em
1
.
Theo em
II
.
Le
k

k
Xi
-
V_

S2n
u , e4n

Yi
=
V
S2n
U9i e4n

i
=
1
.
.. ,
H
.
Assume
ha
i,
g
j
ep esen
elemen s o
in ini e
o de
in 7 4n_
1(V
k
S2n)
i
and
only
i
i
<
h,

j
<
h'
.

Then
i
o
all
i
<
h,
and
H H
V
Xi
-V
Yi
we
ha e
h
=
h'
and
he e
exis
posi i e
in ege s
s,
1,
. . . ,
h
and
a pe -
mu a ion
T
o
{1,
.
. .
,
h}
such
ha
i

i
V
X
i
-
V
Y
,
(¡)
s
H

s
H
/( /
Xi)
-
V(V
h+l h+l
718

I
.
LLERENA
Powe -subs i u ion
A
ing
E
sa is ies
he
"(le )
powe -subs i u ion
p ope y"
i
gi en
xa+
b
=
1
in
E
he e
exis
a
posi i e
in ege
and
a
ma ix
T
E
M (E)
such
ha
I
a+Tb
is
an
uni in
M
(E)
.
He e
I
is
he
iden i y
x
ma ix
.
All
sub ings
o
he
a ionals
Q
and
in
pa icula
7L,
as
well
as
all
he
ma ix
ings
M,,,(ZL),
n
>_
1,
sa is y
powe -subs i u ion
([5,
(2
.9)
and
(3
.4)])
.
So, gi en
XA+B=I
n
in
M
n
(ZZ),
he e
exis
a
posi i e
in ege
and
a
ma ix
T
E
M
n
(ZZ)
such
ha
I
®A+
T
(I
®
B)
is
in e ible
.
He e
®
deno es
he
K onecke
p oduc
A
0
.
. .
0
0
A
.
. .
0
I ®A=
(0
0
. . .
Powe -subs i u ion
implies
powe -cancella ion
in
he
ollowing
sense
.
I
A, B,
C
a e
igh
R-modules
and
End
R
(A) has he
powe -subs i u ion
p ope y,
hen
A®
B
=A®C
implies
B
-
C
o
some
posi i e
in ege
.
P oo
o
Theo em
I
Since
and
g
a e
suspension
elemen s,
hey
ep esen
elemen s
in
®7 -

,_1(Sn)
C
7 ,
n
_
1
(V
k
Sn)
.
So,
and
g
a e
de e minad
by
column
ma ices
x( )
and
x(g)
wi h
en ies
in
7 ,-1(S')
.
I
was
shown
in
[7]
ha
he
mapping
ones
o
and
g,
X
and
Y, a e
in
he
same
genus
i
and
only
i
he e
exis s
an
in ege
ma ix
A
such
ha
x(g)
=
Ax(
)
and
(de
A,
l )
=
l,
whe e
l
deno es
he
o de
o
.
Le
B
be an
in ege
ma ix
such
ha
AB
=
de
A
I
k
,
and
ake
,
s
such ha
de
A
+
sl
=
1
.
Since
7L
has he powe -subs i u ion
p ope y,
he e
exis
an
in ege
c
and
a
ma ix
C
such
ha
I,
de
A+C
1

is
in e ible
.
Then
POWER-CANCELLATION

719
Ik®(I~de A)+(Ik®Cl)=(1,0B)(I,®A)+(Ik®Cl)=V
is
in e ible
and,
by
he
powe -subs i u ion
p ope y
o
M,k(7Z),
we
ob-
ain
an
in ege
d
and
a
ma ix
D
such
ha
Id
0(I~®A)+D(Id®V-1(Ik®Cl))=U
has
an
in e se
.
So
applying
powe -subs i u ion
once
mo e,
we
ob ain
an
in e ible
ma ix
o
he
o m
Q=Ie®(Icd®A)+E(Ie®U
-1
D(Id®V
-1
(Ik0CI)))=I ®A+TI
whe e
=
cde
and
T=
E(I,
®U
-1
D(Id
®
V
-1
(I
k
(D
C)))
.
Now
líen e
P oo
o
Theo em
II
k
Fi s
o
all,
ecall
ha
he
homo opy
ype
o each
X
i
=
V
S2n
U
i
e4n
is
de e mined
by
he
k
xk
in ege
ma ix H( i)
o
he
associa ed
Hil on-
Hop
quad a ic
o m and one one-column
ma ix
x(~

he
en ies o
which
a e
suspension elemen s
in
7 4
n
(S?n
+1
)[1]
.
Mo eo e ,
V
H
Xi
-
Y
i
and
only
i
he e
exis s a
homo opy
commu a i e
diag am
H
V
Son-1
19
H
V
S4n-1

X-V
Y
.
h
. .
.
H
H
k
1
(
S2n)
H
k
91V
.VgH`
V(V
S2n)
=x(gV
.
.
.Vg)
.

72
0

1
.
LLERENA
wi h
d
and
cp
homo opy
equi alences
.
Le
B
i
j be
he
deg ee o
H
H
Son-1 a

V
S4n-1

19

Son-1
~
S4n-1
.
The
ma ix
(Bi )
=
0
cha ac e izes
he
homo opy
class
o
19
.
Le
now
Oil
be
he
ma ix
o
k

H
k

H
k

k
S2n
2nj
->
V(V
S2n)
--W-+
V(V
S2n)
-P i>
V
S2n
.
The
homo opy
commu a i i y
o
he
p e ious
diag am
is
equi alen
o
he
ollowing
condi ion
[1],
[8]
.
O7i
H
( i)oji
=
e7iH(g9)

o
all

i,
j
(*)

OjiH( i)Oli
=
0

i

l
:,A
j
O9ix(1
:
i)
=
e7ix(Egj)

o
all

i,
j
whe e
Oji
is
he
anspose
o Oji
.
We
know
ha
i
is
o
ini e
o de
i
and
only
i
H( i)
=
0
.
Hence,
B
; i
=
0
o j
<
h'
,
i
>
h
.
Thus,
de
0
=
1
implies
h
>_
h'
and
by
symme y
h
=
h'
.
Now
w i e
he
ma ices
o
í9
and
cp
in
he
o m
They
a e
in e ibles
.
Thus
=
C 61
0)

C
4
>1
'P2
/
63
64

_
~D3
~D4
C1-P1+CA3=I
034)2+C4'P4
=
I
whe e

C3

C4

=0
-1
.
By
he
powe -subs i u ion
p ope y
o
Mkh(Z)
and
Mk(H-h)
(7L),
we
can
ind posi i e
in ege s
,
s
and
ma ices R,
S
such
ha
A=
I,
.®<D
1
+R(I,0CA
3
)
B=
I
s
®'D4
+
S(I
s
®
C3~D2)
a e in e ibles
.
Conside
he
diag ams
and
POWER-CANCELLATION

72
1
h

h

(V
i)


h

k
V(V
S
4n-1
)
_
V(V(V
S2n))
V
(V
S
4n-1
)
V

V(V(V
S
2n
))
SH
V(V
S4n-1)
h+l
s
H
h
H
V(V
i)
s
H
h+1
SH
k
V
(V
(V
S2n))
h+l
la
V(V
gi)
V
(
V
S
4n-1
)

V(V
(V
S
2n
))
h+l

h+l
SH
k
whe e
(, ~,
a
and
0
a e
homo opy
equi alen es
wi h
ma ices
I
01
,
I,,
®
194,
A
and
B
espec i ely
.
By
checking
he
ma icial condi-
ions
men ioned
abo e
(*),
we
can
see
ha
hese
diag ams
a e
homo opy
commu a i e
.
(See
[81
o
he
de ails
in
a
qui e
simila
case)
.
So,
hese
diag ams
induce
homo opy
equi alen es
h

h

SH

SH
V(VXi)
=V(VYj)
and
V(V
X
i
)
-V(V
Yi)
h+l h+l
We
may
now
assume
ha
all
he
gi en
spaces
Xi
and
Y¡,

i
=
1,
.
.
. ,
H
,
ha e
a aching
maps
o
in ini e
o de
.

Le
be
he
maxi-
mum
ank
o
he
ma ices
H(
i
)
and
H(gi)
and
assume
ha
he
ank
o

H(g1)
is
.

Since
de
e
=±
l,

0,1
:7~
0,
o
some
l
and,
hence,
01,H( l)oil
=
011H(g1)
has
ank
.
Thus
kH(
l )
=
.
Now,
om
he
es
o
he
ma icial
condi ions
(*),
one
ge s
9jl
=
0
i
j
:~
1,
and
0,1
= l
.
Fo
simplici y
we
suppose
l
=
1
.
Le
us
w i e
he
ma ices
o
99
and
~p
in
he
o m
_

011

C3
)

0
_
(
011

<
1
>
2
0
84

<
D3
'D4
722

I
.
LLERENA
and
le
0
-1
=

Cl

C2

.
Thus
~!3
~!4
0
1011
+
CA3
=
I

and

C3>2
+C
4
-
4
=
I
.
As
be o e,
we
can
ind
in e ible
ma ices o he
o m
A=I,00,1+R(I
®C2'P3)
and
B=I
s
®<D4+S(I
s
®C3)2)
.
Conside
homo opy
commu a i e
diag ams
V
S4n-1
V
S4n-1
s
H
V(V
S4n-1)
2
s
H
V(V
S4n-1)
2

VXl=VY,
and
V
1


k
V
(V
S2n)
s
H
V( 2)
V(V(V
S
2n
))
2
s
H
V( g2)
V(V(V
S2n))
2
10
whe e
(, ~,
a
and,3
a e
homo opy
equi alen es
wi h
ma ices
B11I ,
,
I5
04
,
A
and
B
espec i ely
.
These
diag ams
induce
homo opy
equi a-
lences
s
H

s
H
V(VX
i)=V(VY)
.
2

2
This,
by
induc ion,
comple es
he
p oo
o
Theo em
II
.
Rema k
.
The
ollowing
ques ion
na u ally
a ises
in
he
s udy
o can-

cella ion
:
is
X
-
Y
equi alen
o
V
X
-
V
Y?
(See
[61
o
an
algeb aic
esul
o his
kind)
.
Theo em
I
and
Theo em
1
gi e
a
posi i e
answe
o
some
co-H-spaces wi h
ew
cells
.
Fo
mapping
ones
X
and
Y
o
POWER-CANCELLATION

72
3
elemen s
o
in ini e
o de
in
7 4n_1(VkS2n),
i
ollows
om
[8]
Theo-

em
1
ha
V
X
_
V
Y
implies
X
-Y
.

In
his
case,
howe e ,
he
con e se
is
no
always
ue
.
I ,
o
ins ance,
and
g
a e
such
ha
H( )=
(

0)
,
H(g)
=
(0

2pq)
,
E
=0
and
L
.g=0,
0
2q
whe e p
:,~
q
a e
p ime
in ege s,
hen
X
-Y
and
X
9~
Y
(see
[2])
.
Bu ,
since
H( )
and
H(g)
a e
o
maximum
ank,
i
ollows
om
[8]
Theo em
2

ha
V
X
9~
V
Y
o
all
in ege s
.
Re e ences
1
.

BOKOR,
L,
On
genus
and
cancella ion
in
homo opy,
Is ael
J
.
o
Ma h
.
73
(1991),
361-379
.
2
.

BOKOR,
I
.,
On
he
connec ion
be ween
he
opological
genus
o
ce -
ain
polyhed a
and
he
algeb aic
genus
o hei
Hil on-Hop
quad a ic
o ms,
Publicacions
Ma emá iques
34
(1990),
323-333
.
3
.

BOKOR,
L,
On
mapping
cones
o
suspension elemen s
o
ini e
o de
in
he
homo opy
g oup
o
a
wedge
o
sphe es,
P oc
.
Am
.
Ma h
.
Soc
.
( o
appea )
.
4
.

EVANS,
E
.
G
.,
K ull-Schmid
and
cancella ion
o e
local ings,
Pac
.
J
.
o
Ma h
.
46,
no
.
1
(1973),
115-121
.
5
.

GOODEARL,
K
.
R
.,
Powe -cancella ion
o
g oup
and
modules,
Pac
.
J
.
o
Ma h
.
64,
no
.
2
(1976),
387-411
.
6
.

GURALNICK,
R
.
M
.,
The
genus
o
a
module,
J
.
o
Numbe
Theo y
18
(1984),
169-177
.
7
.

LLERENA,
I
.,
Wedge
cancella ion
and
genus,
P ep in
.
8
.

LLERENA,
I
.,
Wedge
cancella ion
o
ce ain
mapping
cones,
Comp
.
Ma h
.
81
(1992),
1-17
.
9
.

MISLIN,
G
.,
"The
genus
o
an
H-space,"
Lec
.
No es
in
Ma h
.
249,
Sp inge , 1971,
pp
.
75-83
.
10
.
MISLIN,
G
.,
Cancella ion
p ope ies
o
H-spaces,
Comm
.
Ma h
.
Hel
.
49
(1974),
195-200
.