The span and the stable span of a manifold
Abstract
Koschorke, Ulrich
Full text
Pub
.
Ma
.
UAB
Vol
.
26 N2
3
Des
.
1982
THE
SPAN
AND
THE
STABLESPANOF
A MANIFOLD
Ul ich
Koscho ke
Gi en
a
closed
connec ed
smoo h
n-dimensional
mani old
M,
i is
in e es ing
o
compa e
i s
span
(i
.e
.
he
maximum
numbe
o
linea lyindependen angen ec o
ields)
o
i s
s able
span
(i
.e
.
he
maximum
numbe
k
such ha
TM
®
IR
allows
k +
1
linea ly
independen
sec ions)
.
The
example
o
he
s-sphe e
Sn
shows ha hese
wo
numbe s
can
di e d ama i-
cally
:
s able
span
(S
n
)
= n
whilespan
Sn
is
he
Hu wi z-
Radonnumbe
h(n),
e
.g
.
h(n)
=
0
o
n
e en,
h(n)
=
1
o
n
--_
1(4),
h(n)
=
3
o
n
3(8)
. .
.
I
has
been
known
since
long
ha
o
gene al
M
n
o en
span(M)
equals
ei he
he
s áble
span
o
M
o
else
he
span
h(n)
o
he
sphe e
S
n
o
he
same
dimension
(see
e .g
.
[11
o
he
casewhen
M
is
s ably
pa allelisableand
[21
o
manyo he
cases),
so
i
was
widely
belie ed
ha his
shouldalways
hold
.
In
ou
alk
we
disp o e
his
conjec u e
.
Fi s we de ine
an
in ege
s(M)
(%
span(S
n
))
which
can
some imes
be
calcula ed,
e
.g
.
s(M)
=
0
i
n
is
e en
;
s(M)
= 1
i
n
=
1(4)
and
w
l
(M)
2
=
0
;
s(M)
=
2
i
n
-_-
1(4),
w
l
(M)
2
:
A
0
bu
w
1
(M)
2
=
yw
1
(M)
+
y2
o
some
y
E
H1 (M
;
TZ
2
) ;
e c
.
Then
we
use
he
singula i y
app oach
o
ec o ield
p oblems
(see [31)
o
show
ha mos
o
he
ime
s(M)
is
he
co ec al e na i e
:
he
span
o
M
is
equalei he o
s(M)
o o
he
s able
span o
M
.
As an
example
we
exhibi
an in ini e amilyo mani olds
o
he
o m
Mn = P x
Sq
such
ha
span(M)
di e s
om
s able
span
(M),
10
7
om
span
(S
n
)
and
also
om
he
sean
o
he
ac o sphe e
Sq,
bu
span(M)
=
s(M)
=
4
.
We
gi e
many
o he
coun e exam-
ples
o
he
conjec u e
men ioned
abo e,
including
o ien ed
ones
.
Re e ences
(Mo e
de ails
a e
gi en
in
he
las
sec iono
[3])
.
[11
G
.
B edon
and
A
.
Kosinski,
Vec o ields
on
u-mani olds,
Ann
.
Ma
.
84
(1966),
85-90
.
[21
V
.
Eagle,
Ru ge s
Ph
.
D
.
hesis,
1978,
(unpublished)
.
[31
U
.
Koscho ke
Vec o ields
and
o he ec o
bundle
mo phisms
-a
singula i-
y app oách,Lec
.
No es
in
Ma h
.,
ol
.
847,
Sp inge -Ve lag,
1981
.
Uni e si á
-
Gesam hochschule
-
Sieaen
Deu schland
10
8