Publicacions
Ma emá iques,
Vol 35
(1991),
527-535
.
APPLICATIONSOF
THE
EULER
CHARACTERISTIC
IN
BIFURCATION
THEORY
Abs ac
SLAWOMIR
RYBICKI
Le
:
R"
x
R
k
-
.
R"
be a
con inuous
map
such
ha
(O,, )
=
0
o all
AE
Rk
.
In
his
a icle
we
o mula e,
in
e ms
o
he
Eule
cha ac e is ic
o
algeb aic
se s,
su icien
condi ions
o
he
exis en e
o
bi u ca ion
poin s
o
he
equa ion
(x, A)
=
0
.
Mo eo e
we
apply
hese
esul s
in
bi u ca ion
heo y
o
o dina y
di e en ial
equa ions
.
I is
wo h
o
poin
ou
ha
in
he
las
pa ag aph
we
show
how
o
e i y,
by compu e ,
he
assump ions
o
he
heo ems
o
his
pape
.
0
.
In oduc ion
In [A]
Alexande has
de ined
an
in a ian
which
non i iali y
implies
he
exis en e
o a
bi u ca ion
poin
o a
con inuous
map
:
R'
x
R
k -->
R'
such
ha
(0,
A)
=
0
o
all
A E
R
k
.
This
in a ian
is
an
elemen
o
he
g oup
7 k_1(GL(n))
.
Gene ally
i is
di icul
o
e i y
i
his
in a ian
is
a
non i ial
elemen
in
7 k-I(GL(n))
.
K asnosielski
in
[K] has
p o ed
a
1-pa ame e
bi u ca ion
heo emwhich
is
a
e y
use ul
ool
in
bi u ca ion
heo y
.
This
heo em
gi es
su icien
condi ions
o
he
exis en e
o
a
bi u ca ion
poin
o
in
he
case
k
=
1
.
Many
au ho s
ha e
p o ed
gene aliza ions
o
he
classical
K asnosielski
he-
o em
(see
[MA],
[R1], [R2])
.
We
a e
in e es ed
in
o mula ing
su icien
condi ions
o
he
exis ence
.o
bi u ca ion poin s
o
in
case
he
dimension
o
he
pa ame e
space
is
g ea he
han
one
and
when
he
Alexande
in a ian
can
no
be
applied
.
In
[S1],
[S2]
and [W]
he
au ho s
Na e
p o ed
e y
in e es ing
o mulas o
a
compu a ion
o
he
Eule
cha ac e is ic o
algeb aic
se s
in
e ms
o
he
B ouwe
opological
deg ee
o sui able
maps
.
In
he
i s
pa
o
his
pape
(using
hese o mulas)
we
o mula e
and
p o e
su icien
condi ions
o
he
exis en e
o
bi u ca ion
poin s
o
.
Namely,
we
de ine
(De
.
1
.4
.)
a
se
o
essen ial
maps
(Ess
(n, k))
and
p o e
ha
0
-E
R
is
no
an
isola ed
bi u ca ion
poin
o
any
essen ial
map
(P op
.
1 .1
.)
.
52
8

S
.
RYBICKI
We
also de ine
(De
.
1
.3
.)
a
se
o
egula
maps
(Reg
(n,
k))
and
show
how
o
e i y
i
a
egula
map
is
an
essen ial
map
(P op
.
1
.3
.)
.
In
P oposi ion
1
.4
.
we
o mula e
su icien
condi ion
o
he
exis ence
o
a
bi u ca ion
poin
o
a
homogeneous
map
( o
de ini ion
o a
homogeneous
map,
see
De
.
1
.2
.)
.
As
he
las
case
we
conside
a
se
o
e en
maps
(E en
(n, k),
De
.
1
.5
.)
.
In
P oposi ion
1 .5
.
we
gi e
su icien
condi ions
o
he
exis ence
o a
bi u ca ion
poin
o
e en
map
.
No ice
ha
he
assump ions
o
P oposi ions
1
.4
.,
1
.5
.,
1
.6
.
a e
exp essed
in
e ms
o
he
B ouwe
opological
deg ee
o
polynomial
maps
.
F om
his
poin
o
iew
i is
impo an
o
compu e
he
B ouwe
opological
deg ee
o
polynomial
maps
.
Nie enbe g
has
o mula ed
in
[N]
án
in eg al
de ini ion
o
he
B ouwe
opo-
logical
deg ee
.
.
We
ha e
w i en
a
compu e
p og am whichcompu es
he
opo-
logical
deg ee
o
polynomial
maps,
in
a
e sion
gi en
by
Nie enbe g
.
The
impo an
ques ion
is
how
o
e i y
ha
E
Reg
(n,
k)
.
In
o he
wo ds
we mus
e i y
i
0
E
Rk
is
an
isola ed
poin
in
IP
-1
(0)
.
The e
a e
compu e
algo i hms
which
allow
us
o
check
i
0
E
R
k
is
an
isola ed
ze o o
he
polynomial
map,
T
:
R
k -->
R
k
.
The e
algo i hms
a e
based
on
he
Eisenbud
and
Le ine
esul s
(see
[E.L
.])
.
The e
is
a
compu e
p og am,
w i en by
And zej
Lecki
om
Uni e si y
o
Gdansk,
which
is
based
on
such
kinds
o
algo i hms
.
Using
his
p og am
we
can
e i y
i
E
Reg
(n,
k)
.
Acknowledgemen
.
The
au ho wishes
o
hanks
o
And zej
Lecki
o
se e al
help ul
commen s
.
In
he second
pa ag aph
we
apply
opological
esul s o
his
a icle
o
he
bi u ca ion
heo y
o
o dina y
di e en ial
equa ions
(Th
.
2
.1
.,
2
.2
.,
2
.3
.)
.
In
he
las
pa
o
his
pape
we
show
how
o
e i y
by
compu e
he
assump-
ions o
he
heo ems
o his
pápe
.
1
.
Resul s
Deno e by
X
and
Y
Banach
spaces
and by

X
x
R
k
->
Y
a
con inuous
ope a o such
ha
(0,
A)
=
0
o
all
a
E
R
k
.
De ini ion
1 .1
.
A
poin
ao
E
R
k is
said o
be
a
bi u ca ion
poin
o
he
equa ion
(*)
(x,, )
=
0,
i
(0,
o)
E

closu e
{(x,
. )
E
X
x
R
k
:
(x,
.1)
=
0and x
:,A
0}
.
The
se
o
bi u ca ion
poin s
ó
he
equa ion
(*)
will
be
deno ed
by
Bi ( )
.
Conside
a
C
1
-map
:
R'
x
R
k
-
>
R'
such ha
(0,
.~)
=
0
o
all
A E
R
and
de ine
a
map
(D
:
R
k
->
R
by
~P
(A)
=
de (D
.,
(0,
A))
.
De ini ion
1 .2
.
A
map
is
said o
be
a
homogeneous map,
i
-P(A)
is
a
homogeneous
polynomial
o
deg ee
g ea he
han
1
.
APPLICATIONS
OF
THE
EULER
CHARACTERISTIC

529
The
se
o
homogeneous
maps
will
be
deno ed
by
Hom
(n,
k)
.
De ini ion
1 .3
.
A
map
:
R"
x
R
k
->
R'
is
said o
be
a
egula
map,
i
1)
E
Hom
(n,
k),
2)
<D
has
an
isola ed
c i ical
poin
a
he
o igin
.
The
se
o
egula
maps
will
be
deno ed
by
Reg
(n,
k)
.
De ine
a
map
IP
:
R
k
-->
R
k
by
he
o mula
2
+
.p(a)
2
,. . . ,
La
4)
(A)]
2
+
~(a)2
and
no ice
ha
i
T(Ao)
=
0,
hen
xP(
-
Xo)
=
0
o
all
E
R
.
Rema k
1
.1
.
The
map
~¿
has
an
isola ed
c i ical
poin
a
he
o igin
i
I
P-1(0)
=
{0}
.
Using
his
ema k
we
will
show
ha
a
map
conside ed
in
Example
3
.2
.
is
a
egula
map
.
De ini ion
1 .4
.
Amap
:
Rn
x
R
k
->
Rn
is
said
o
be
an
essen ial
map,
i
1)
E
Reg
(n, k),
2)
P-1(0)
=,A
{
0
}
.
The
se
o
essen ial
maps
will
be
deno ed
by
Ess
(n,
k)
.
P oposi ion
1 .1
.
I
E
Ess
(n,
k),
hen
Bi
( )
=
~D
-1
(0)
.
In
.pa icula ,
Bi
( )
=~
0
.
P oo
:
:
The
se
<I)
-1
(0)
-
{0}
is
a (k
-
1)-dimensional
mani old
.
Fix
Ao
E
ob-1(0)-{0}
and
1-dimensional mani old
N(A0)
such
ha
mani olds
N(Ao)
and
(D-1(0)-{0}
a e
ans e sal
a
he
poin
Ao
.
I
is
easy
o see
ha
he
map
<D
changes
sign,
a
he
poin
Ao,
along
N(Ao)
.
So,
applying
K asnosielski
heo em
(see
[K])
we
show
ha
Ao
E
Bi
( )
.
The
se
Bi
( )
is
closed,
ha
is
why0eBi ( )
.
a
Deno e by
d(D
:
R
k
-
R
k
he
g adien
o
he
map
I¿
.
I
E
Reg
(n,
k)
hen
dD
-1
(0)
=
{0}
.
Deno e
by
X(B)
he
Eule
cha ac e -
is ic
o
he
mani old
B
and by
deg
H
he
B ouwe
opological
deg ee
o
d(D
wi h
espec
o
a
disc
cen e ed
a
he
o igin
.
P oposi ion
1
.2
.
I
E
Ess
(n,
k),
hen
o
any
E
> 0
i)
D-1(0)
n
SÉ
-1
is
a
(k
-
2)-dimensional
mani old,
ü)
X
(
,
P
-1(0)
n
SE
-1
)
_
(1
+
(
-1
)
k
)
'
(
1
-
degd<D)
.
P oo
::
No ice
ha
<P
-1
(0)-{0}
is
a
(k-1)-dimensional mani old
ans e sal
o
SÉ
-1
.
Tha
is
why
<I>
-
'(O)
n
SÉ
-1
is
a
(k
-
2)-dimensional
mani old
.
Le
us
pu he
ollowing
no a ions
A
+
_
{A
E
SÉ
-1
:
ID(A)
>_
0},
A_
_
{A
E
SÉ
-1
:
-P(A)
<
0},
L=
{R
E
SÉ
-1
:
-P(A)
=
0}
.

,
53
0

S
.
RYBICKI
C
.T .C
.
Wall
in
[W]
has
shown
ha
X(A+)
=
1
+
(-1)
k-1
-
degdP
and
x(A_)
=
1
-
deg
d-D
.
I is
well
known
ha
X(L)
=
x(A
+
)
+
X(A_)
-
X(Sk-1)
.
Using
he
abo e
o mula
and
C
.T
.C
.
Wall
esul s
we
ob ain
he
hesis
.
P oposi ion
1
.3
.
I
E
Reg
(n,
k)
and
(1+
(-1)
k
)
-
(1-deg
dD)
7~
0,
hen
E
Ess
(n,
k)
.
P oo
.-
I
E
Reg
(n,
k),
hen
~P
-1
(0)nSÉ
-1
is
a
(k-2)-dimensional
mani old
o
is
an
emp y
se
.
F om
P oposi ion
1
.2
.
and
om
he
assump ions
i
ollows
ha
~¿
-1
(0)
n
SÉ -1
is
no
emp y
se
.
So
ou
p oo
is
comple ed
.
In
his
pa
o
he
pa ag aph
we
o mula e
su icien
condi ions
o
he
exis-
ence
o
bi u ca ion
poin s, in
a
case
when
E
Hom
(n,
k)
.
Le
us deno e
B
i
=
{A
E
Si
-1
:
(-1)
i
+
1
-
-D(A)
>_
0}
o
i
=
1,
2
and
le
m
be
he
deg ee o
he
polynomial
P(A)
.
I
m
is
an odd
numbe
hen
ob
has
o
change
a
sign
nea
he
o igin
ha
is
why
i is
enough
o
conside
only
he
case
o
e en
m
.
De ine polynomials
Fi
:
Rk
->
R
by
he
o mula
o
i
=
1,
2
.
By
d i
:
R
k
,
R
k
we
deno e
he
g adien
o
Fi
.
By
deg
(Q,
DÉ
,
0)
we
will
deno e
he
B ouwe
opological
deg ee o
he
map
Q
compu ed
on
he
disc
DÉ
wi h
espec
o
he
o igin
.
P oposi ion
1 .4
.

The e
is
a
posi i e
numbe
E
>
0 such
ha
dF~
1
(0)
n
DÉ
_
{0}
.

I
deg
(dFi,DÉ,0)
~
(-1)
k
o
i
=
1, 2,
hen o
any
el
>
0 Bi
( )
n5,
k
,
-1
=,A
0,
in
pa icula ,
0
E
Bi
( )
.
Mo eo e
he
opological
dimension
o
he
se
Bi
( )
n
5,
k
,
-1
is
equal
o
k
-
2
.
P oo
. .
Fi s o
all
we
will
show
ha
X(Bi)
7~
X
(Sk
-1
)
o
i
=
1,
2
.
Sza aniec
has
shown
in
[S2]
ha
he e
is
s

>

0
such
ha
dFZ
1
(0)
n
DÉ

=

{0}
.
I
deg
(dFi,
Dk
,
0)
=~ (
-
1) k
,
hen
1
-
deg(dF
i ,
DÉ
,
0)
5
E
1
+
(-1)k
-1
.

I
is
well
known
ac
ha
X(S
k-1
)
=
1
+
(-1)
k-1
.
F om
Co olla y
1
in
[S2]
i
ol-
lows
ha
X(Bi)
=
1-deg(dF
i
,
DÉ,
0)
.
So,
we
ha e
x(Bi)

X(Sk
-1
)
.
F om
he
abo e
i
ollows
ha
Bi

S
k-1
o
i
=
1,
2
.
Fix
Ai
E
Sk-1
-
Bi
o
i
=
1,
2
.
Le
[0,1]
->
S
k-1
deno e
a
con inuous
map
such
ha
l
;'(0)
_
A1
and
j(1)
=
A2
.
Conside
a
Pmposi ion
<P
o 1
:
[0,1]
-->
R
.
I is
ob ious
ha
(~P
o
)(~1)
(4>
o
1)
(A2)
<
0
.
So,
om
K asnosielski
heo em
(see [K])
i
ollows
ha
Bi
(
)
n
«[0,1])
=,A
Q)
.
I
is
easy
o
e i y
ha
i
Ao
E
Bi
( ),
hen
-
Ao
E
Bi ( )
o
all
E
R
.
So,
ou
p oo
is
comple ed
.
In
he
las
pa
o
his
sec ion
we
u n o
a
case
when
a
map
is
such
ha
d>(A)
is
no necessa y
an homogeneous
polynomial
.
De ini ion
1 .5
.
A
map
:
R'E
x
R
k
->
R"
,
is
said o
be
an
e en
map,
i
-D(A)
is
a
polynomial
such
ha
4)(-A)
=
The
se
o e en
maps
will
be
deno ed
by E en
(n,
k)
.
Fix
E
E en
(n,
k)
and
deno e
by
d
he
deg eee
o
<>
.
Choose
any
e
>
0
.
De ine
a
map
A,
:
DÉ+I
-,
R
as ollows
Rema k
1
.2
.

Nó ice
ha
A,
(A)
is
ahomogeneous
polynomial
and
ha
sgn
A,
(A)
=
sgn
D(P(A))
o
A
E
SÉ
,
whe e
P
:
SÉ
->
DÉ
is
he p ojec ion
gi en
by
he
o mula
P(A)
=
(al,
. . . ,
k)
.
De ine
polynomials
Ei
:
Rk+I
->
R
as ollows
o
i
=
1,2
.
By
d
_'
:
Rk+I
-->
Rk+I
we
will
deno e he
g adien
o
Es,
P oposi ion

1 .5
.
Fo
su jicien ly
small
El

>

0,

i
deg
(d
E',
D
k
1
,
0)
(-1)k+
1
o
i
=
1, 2,
hen
Bi
( )
n
D
k

:,A
0
;
mo eo e
he
opological
dimension
o
he se
Bi
( )
n
D
k

is
equal
o
k
-
1
.
A
p oo
o
his
p oposi ion
is
a
consequence
o
P oposi ion
1
.4
.
and
Rema k
1
.2
.
In
his
sec ion
we
will
use
he
no a ions
o
he
i s
sec ion
.
Conside
a
C
l
-map
g
:
R
x
R'
x
R
k
-+
Rn
and
assume
ha
g( ,
x,
A)
can
be
exp essed
in
he
o m
whe e
APPLICATIONS
OF
THE
EULER
CHARACTERISTIC

531
.
_1
AE(a)
=AE(a1,
. .
.,ak+I)
=
II~IId+2

CP 11
Eñl
' .
.''
II~1I
EA
k
J
L i
(A)
_
(-1)'AE(A)
-
(al
+

. .
+
ñk+I)`~'+I
2
.
Applica ions
g( ,
x, A)
=
A(A)x
+
(p( ,
x,
A),
1)
A(A)
is
a n x
n-ma ix
such
ha
A(0)
=
0,
2)

W( ,
0,
A)
=
0
o
all
( ,
0,
A)
E
R
x
Rn
x
R
k
,
3)
D
x
cp( ,
0,
A)
=
0
o
all
( ,
0,
A)
E
R
x
Rn
x
R
k
.
We
a e
in e es ed
in
desc ibing
he
se
o
bi u ca ion poin s
o
he
ollowing
bounda y
alue
p oblem
(*)

x
( )
=
g
( ,
x
( ),
~)
x(0)
=
x(1)
.

532

S
.
RYBICKI
Le
us
deno e
X
=
{x
E
C
l
([0,1])
:
x(0)
=
x(1)}
and
Y=C
o
([0,1])
.
De ine
an
ope a o
F
:
X
x
Rk
->
Y
by
he
o mula
F(x( ),
A)
=
L(A)(x( ))
-
cp( ,
x( ),
A),
whe e
L(A)(x( ))
=
x( )
-
A(A)x( )
.
No ice
ha
ze oes
o
he
ope a o
a e
in
one- o-one
co espondence
wi h
solu ions o
he
p oblem
(*)
.
Le
:
R'
x
R
k
--->
R''
be
he
map
de ined
by
(x,
A)
=
A(A)x
.
Now
we
a e
in
a
posi ion
o
o mula e
he
main
heo em
o
ou
pape
.
Theo em
2
.1
.
I
E
Ess
(n,
k)
and
E
is
a
su ccien ly
small
posi i e
numóe ,
hen
1)
Bi (F)
n
DÉ
=
4
-1
(0)
n
DÉ
,
2)
Bi (F)
n
SÉ
-1
is
a
(k
-
2)-dimensional
mani old,
3)
x(Bi
(F)
n
SÉ
-1
)
=
x(D
-1
(0)
n
SE
-1
)
=
(1
+
(-1)
k
)
.
(1-
degdD)
.
P oo
.
No ice
ha
L(0)
:
X
-->
Y
de ined
by
L(0)(x( ))
=
x( )
is
a F edholm
ope a o
wi h
F edholm
index
0 and
ha
X
=
Xo
®
Xl
and
Y
=
Yo
®
Y,,
whe e
Xo=
ke
L(0)
=
R
n
=
{subspace
o cons an
unc ions},
(see
[M]
o
mo e
de ails)
.
Yo
=
R'
=
{subspace
o
cons an
unc ions},
1
Yl
=
im
L(0)
=
{x
E
C
o
[0,1]
:
x(s)
ds
=
0},
0
We
begin
wi h
he
Lyapuno -Schmid
educ ion
.
Le
Po(x)
=
o
x(s)
ds
and
P,
(x)
=
x
-
ó
x(s)
ds deno e
he
p ojec ions
o
Y
on o
Yo,
Y,,
espec i ely
.
Then
he
equa ion
F(x,
A)
=
0
is
equi alen
o
he
sys em
o
equa ions
Po(F(xo
+
xl,
A))
=
0,
P,(F(xo
+
x1,A))
=
0,
whe e
x
=
xo
+x1,
xo
E
Xo,
-xl
E
Xl
.
No ice ha
he
map
P
l
oF
:
Xo
®Xl
®R
k
--+Y,
is
con inuously
di e en iable
nea
(0, 0,
0)
E
Xo
®
Xl
®
R
k
,
P
l
oF(0,
0,, )
=
0,
and
he
FYéche
de i a i e
o Pl
o
F
wi h
espec
o
xl
a
(0,
0,
0),
D
xl
Pl o F(0,
0,
0)
is
an
isomo phism
o
Xl
on o
Y,
.
The e o e
by
he
implici
unc ion
heo em,
he e
is
an upen
neighbou hood
U
o
(0,
0)
E
Xo
®R
k
and
xl
E
C
l
(U,
X
I
)
such
ha
he
ze os
o
F
nea
(0,
0,
0)
a e
gi en
by
(xo,
xl
(xo,
A),
A)
o
(xo,
A)
E
Xo
®
Rk
.
I
is
easy
o see
ha
xl(xo,
A)
=
0(Ilxo11)
a
xo
=
0,
uni o mly
o
A
nea
0
.
APPLICATIONS
OF
THE
EULER
CHARACTERISTIC

53
3
F om
he
abo e
i
ollows
ha
ze os o
F
a e
in
one- o-one
co espondence
wi h
ze os
o
a
ini e
dimensional
map
Q
:
U
-+ Yo
de ined
by
he
o mula
Q(xo
,
A)
=
(Po
-
F)
(xo
+
xl
(xo,
A)
,
A)
.
I
is
no
di icul
o
e i y
ha
Q(xo,
A)
is
a
map
o
he
o m
Q(xo,
A)
=
-
A(A)x0
+T
(xo,
a),
whe e
T(0,
A)
=
0
and
D,,T(0,
A)
=
0
.
No ice
ha
Q
E
Ess
(n,
k)
.
The
es
is
a
consequence
o
P oposi ions
1
.1
.
and
1 .2
.
The
nex
heo ems
gi e
su icien
condi ions
o
he
exis en e
o
bi u ca ion
poin s
o
he
ope a o
F
o
mo e
gene al
class
maps
han
he
class
Ess
(n,
k)
.
Theo em
2
.2
.
I
E
Hom
(n,
k)
and
deg
(d i,
DÉ,
0)
~
(-1)k o
i
=
1,
2,
hen
o
any
el
G
e
Bi
(F)
1
SÉ
-
,
1
0
.
In
pa icula ,
0
E
Bi (F)
.
Mo eo e
he
opological
dimension
o
he
se
Bi
(F)
n
SÉ
;
l
is
equal
o
k
-
2
.
This
heo em
is
a
consequence
o
P oposi ion
1
.4
.
A
p oo
o
his
heo em
is
simila
o
he
p oo
o
Theo em
2
.1
.
Theo em
2
.3
.
Le
E
E en
(n,
k)
.

Then
o
su icien ly
small
E
i
deg
(d Es,
DÉ+1,
0)

(-1)k+1
o
i
=
1, 2,
hen
Bi
(F)
n
D
Ek
0
.

Mo e-
¡
o e
he
opological
dimension
o
he
se
Bi
(F)
n
DÉ
is
equal
o
k
-
1
;
in
pa icula ,
0
E
Bi
( )
.
This
heo em
is
a
consequence
o
P oposi ion
1
.5
.
A
p oo
o
his
heo em
is
simila o
he
p oo
o
Theo em
2
.1
.
3
.
Examples
We
shall
emind
well
known
esul s
om
singula i y
heo y
(see
[E .L
.])
.
Le
0,,
be
he
ing o
ge ms
o eal
analy ic
unc ions
a
0
E
R'
.
I
g1,
. . .
.
gn
E
B
n
,
le
us
deno e
by
(gl,
. . . .
g

,)
he
ideal
in
9

,
gene a ed
by
he
elemen s
91,
. . . .
g
n
.
Fo
g
=
(91,
. . . .
g
n
)
:
(Rn,
0)
--,
(Rn,
0)
we
pu
Q(g)
=e
.M1,
-. . ,
gn)-
I
g
is
ini e,
in
he
sense
ha
Q(g)
is
ini e
dimensional
eal
ec o
space,
hen
0
E
Rn
is
isola ed
in
g-1
(0)
.
Conside
he
ollowing
bounda y
alue
p oblem
:
( )
=
g
( ,
x( ),
x(0)
=
x(1)
53
4

S
.
RYBICKI
and assume
ha
he
map
g
sa is ies
all
he
assump ions
o
he
p e ious
pa a-
g aph
.
So
we
can
exp ess
g
in
he
o m
g( , x,
A)
=
A(A)x+W( ,
x,
A)
.
Theo ems
2
.1
.,
2
.2
.,
2.3
.
show
ha
i is
enough
o
examine
only
he
ma ix
A(A)
.
Example
3
.1
.
Assume
ha
n=
k
=
3
and
ha
he
ma ix A(A)
is
o
he
o m
A1
+
A2,

A2
+
A3,

A
1
+
A3
A(A)
=

A
1
+
A
2
+
A3,

A
l
-
A2,

2
-A
l
-
3
-
A3
A3,

A1,

0
Using
a
compu e
p og am
we show
ha
deg
(d i,
D3,0)
=
1
o
i
=
1,
2
.
F om
his
i
ollows
ha
deg
(d i,
DÉ,
0)
A
(
-
1)
3
,
so
he
assump ion
o
The-
o em
2
.2
.
is
ul illed
.
Applying
Theo em
2
.2
we
claim
ha
he e
exis s
e
>
0
such
ha
o
any
el
<
e
he
in e sec ion
o
he
se
o
bi u ca ion poin s
o
he
p oblem
(**)
wi h
SÉ,
is
no
emp y,
in
pa icula
0
E
Bi
(F)
.
Mo eo e
he
opological
dimension
o his in e sec ion
is
equal
o
1
.
Example
3
.2
.
Assume
ha
n
=
k
=
2
and
ha
he
ma ix A(A)
is
o
he
o m
_

A
1
-
A2,

Ai
-
A2
A(A)
-
[
-A1
+
2
-
A
2
,

A
1
1
A
2
-
A2
]
.
De ine
a
map
:
R
2
x
R
2
->
R
2
by
(x,
A)
=
A(A)x
.
We
will
show
ha
E
Ess
(2,2)
.
Using
a
compu e
p og am
we
show
ha
dim
Q(YP)
=
16,
so
om
he
esul s
om
singula i y
heo y
i
ollows
ha
T
-1
(0)
=
{0},
ha
is
why
E
Reg
(2,2)
.
Compu ing
he
B ouwe
opological
deg ee
(by
compu e )
we
ob ain
deg
d<D
_
-2
and
consequen ly
X(D
-1
(0)
l
SÉ)
=
(1
+
(-1)k)
-
(1
-
degd<D)
=
(1
+
(-1)2)
.
(1
-
(-2))
=
6
M-om
his
i
ollows
ha
4)
-1
(0)
=~
{0},
so
E
Ess
(2,2)
.
So
om
Theo em
2
.1 i
ollows
ha
Bi (F)
(1
D
2
consis o
exac ly
six in e -
als,
which
emana e
om
he
o igin
.
Re e ences
[A]
J
.C
.
ALEXANDER,
Bi u ca ion
o
ze os
o
pa ame ized
unc ions,
J
.
Fune
.
Analy
.
29
(1978),
37-53
.
[E .L
.]
D
.
EISENBUD,
H
.I
.
LEVINE,
An
algeb aic
o mula
o
he deg ee
o
a
C°°
map
ge m,
Ann
.
Ma h
.
106
(1977),
19-44
.
[I
.R
.1]
M
.
IZYDOREK,
S
.
RYBICKI,
On
he
numbe
o
bi u ca ion
b anches
o
C
2
-maps,
accep ed
o
publica ion
in
JMAA
(1991)
.
[I
.R
.2]
M
.
IZYDOREK,
S
.
RYBICIQ,
On
he
numbe
o
b anches
o
bi u ca ion
poin s, o
appea
.
APPLICATIONS
OF
THE
EULER
CHARACTERISTIC

53
5
[K]
M
.A
.
KRASNOSIELSKI,
"Topological
me hods
in
he heo y
o nonlinea
in eg al
equa ions,"
Pe gamon,
1964
.
[M]
J
.
MAWHIN,
"Topological
deg ee
me hods
in
nonlinea
bounda y
alue
p oblems,"
Con e ence
boa d
o
he
ma hema ical
sciences,
Regional
con-
e ence
se ies
in
ma hema ics
40,
Ame ican
Ma hema ical
Socie y,
P o i-
dence,
Rhode
Island,
1979
.
[MA]
A
.
MARINO,
"La
bi u cazione
nel
caso
a iazionale,"
Con
.
Sem
.
Ma
.
Uni
.
Ba
¡
132, 14
pp
.
MR50,
1068,
1973
.
[N]
L
.
NIERENBERC,
"Topics
in
nonlinea
analysis,
N
.
Y
.
Y
.,"
Lec u e
No es,
1974
.
[Rl]
P
.
RABINOWITZ,
Some
aspec s
o
nonlinea eigen alue
p oblems,
Rocky
Moun
.
J
.
o
Ma h
.
3
(1973),
161-202
.
[R2]
R
.
RABINOWITZ,
A
bi u ca ion
heo em
o
po en ial
ope a o s,
J
.
Func
.
Analy
.
26
(1977),
48-67
.
[S1]
Z
.
SZAFRANIEC,
On
he
Eule
cha ac e is ic o
analy ic
and
algeb aic
se s,
Topology
25,
4
(1986),
411-414
.
[S2]
Z
.
SZAFRANIEC,
The
Eule
cha ac e is ic
o
algeb aic
comple e
in e sec-
ions,
J
.
Reine
Angew
.
Ma h
.
397
(1989),
194-201
.
[W]
C
.T
.C
.
WALL,
Topological
in a iance
o
he
Mi1no
numbe
mod
2,
Topology
22
(1989),
345-350
.
Depa men
o
Ma hema ics
Technical
Uni e si y
o
Gdansk
ul
.
Majakowskiego
11/12
80-952
Gdansk
POLAND
P ime a
e sió
ebuda
el
2
d'Oc ub e
de
1990,
da e a
e sió
ebuda
el
3
de
Juny
de
1991