Pub
.
Ma
.
UAB
N
°
21
Oc
.
1980
Ac es
VII
JMHL
FLAT
SUBMERSIONS
AND
HORIZON
IMMERSIONS
F
.J
.
C a ei o
de
Ca alho
Dp o
.
d
e
Ma emá ica
Uni e sidade
de
Coimb a
Abs ac
The no ion
o la
subme sion
is
in oduced
.
We
p o e
ha
no
co e ing
map,
wi h
o de
g ea e han
1,
is
1- la , ela e 2- la
sukene sions
M
n+l
->S
n
o
ho izon
imme sions
and
show ha
k- la

suJxm sions
Mm
_
Sn gi e
ise
o
embeddings
~
n
-
Rpn+k
.
1
.
In oduc ion
Th oughou
his no e
all he
maní oldsa e
C
,
Hausdo ,
compac ,
connec ed
and
bounda yless
.
All
he
maps
a e
C
,
unlesso he wise
s a ed
.
I
M
and
N
a e
mani olds,
o
dimensions
m
and
n
espec i ely,
we
shall
deno e hem
byMm

and

N
n
.
Le

11 :
N?
--+
N
n
be
a
subme sion
.
De ini ion
1
:
We
say
ha
II
is a
k- la
subme sion
(k>m-n)
i
he e
exis s
a
map
F
:
M
m
-o
Rk

such
ha ,
o
e e y
y e Nn
,

F111
-1
(Y)

is
an
embedding
.
P oposi ion
1
:
I
11
is a

k- la
subme sion
hen

( í,F)
:Mm
--N
n
x
Rk

is
an
embedding
.
P oo
:
I
is
clea
ha (1I,F)
is
C~
and
injec i e
.
The
only
hing
o
wo y
abou
is
he
ank
o

(11,F)
.
Le
x
be
any
poin
in
M
m
.
We
p o e
ha
he
ke nel
o

(I1,F)*x
:
Tx
J
--
TII(x)
Nn
x
T
F(x)
R
k

is

O-dimensional
.
As
ke (1T,F)
*x
=
=
ke
R*
X
lke
F
*x
,
i

e
ke (TI,F)
*x
we
ha e

F
*x
( )
=
0
.
On
he
o he
hand
*
AMS

subjec
classi ica ion

57 D
40
**
Suppo ed
in pa
by
Calous e
GulbenkianFounda ion
and
INIC,
Lisbon,Po ugal
eke R*x,ke n
*x = i
*x
(T
x
n-1(R(x))),whe e

i

is
he
inclusion

n
1
( i(x))
-->
M,
and

FIT
_1
(x)

is
an
embedding,
he e o e


mus
be ze o
.
2
.
Fla co e ingmaps
In
his
sec ion
we
deal
wi h
he
case

m=n
.
We
shall
conside
a
,mo e
gene al
si ua ion
.
Le
M
be
a
compac ,
pa h-connec ed
opological
space
and
II
:h
->
N

be a
co e ing
map,
whe e

D
:

is
also
a
opological
space
.
P oposi ion
2
:
Le

n
:M
^
.
A'

be
a
co e ing
map
such
ha ,
o
any

ycN, :W n
-1
(y)
>
1
.
I

:M
-->
R

is a
con inuous
map
hen
he e
exis

x

and

y

in
he
same
ib e
such
ha

(x)

=
(y)
.
P oo
:
We
use
some
ac s
abou co e ing
spaces
.
E4]
.
Le
x
l
be an
absolu e
maximum
and

y
l
be
an
absolu e
minimum
o
.
Take
a
con inuous
pa h
y
:
E0,11
-+
M

such
ha

y(O)
=
x
1
and

y(1)
=
y
1
.Choose

x
2
~xl
,
wi h
l(x
2
)
=
i(x
1
)
and
conside
he
li y' o
POy
such ha
y'(0)
=x2
.
We
de ine
he map

~
:
E0,11
->
R

by aking

O( )

o be

oy( )
-
oy'( )
.
I
0(0)
o
~(1)
is'ze o
hen
he
p oposi ion
is
p o ed
.
I
no ,

mus ha e
a
ze o
O
,
because
~(0)
>
0

and
.
0(l)<
0,
and he
p oposi ion
ollows
.
An
immedia e
consequence
is
Co olla y
1
:
I

:S
n
--
"
R

is
con inuous
hen
he e
exis s

x
ES
n

such
ha
(x)
=
(-x)
.
O
cou seco olla y
1
can
also
be
,
deduced
om
he
Bo s
u
k-Ulam
heo em
L4-
.
3
.
Fla
subme sions
and
ho izon
imme sions
We
s a
wi h
he
de ini ion
o
ho izon
imme sion
L2]
.
Le
:M
-->
Rm+1
be
an
imme sion
and
L
deno e
an a ine line
in
Rm+1
.
De ini ion
2
:
We
say
ha

:b? n
-->
R
m+1

is
a
ho izon
imme sion
wi h
base-line
L i ,
o
e e y
xEJ,
he
a ine
angen
space
o

(M)
a

(x)
does
no
con ain
L
.
In
wha
ollows
we
shall
be
conside ingsubme sions
n
:Mn+l
-}
Sn
and
we
shall
use
he
ollowing
chain
o
di eomo phisms
whe e
X1
(x,
l
,
2
)
=
(x
e
,
2
),
X 2
(x,
l
,
2
)
=(
i
x,
2
)
and
R
deno es
he
posi i e
eals
.
The inclusion

i
:R
n+I
{0} x
R -+
Rn+2

will also
be
used
and
he
composi ion
io X
2
o X
1
will be
deno ed
by
X
.
F om
now
on
L
will
deno e
he

a ine

line

{
(0,
. .
.
,0,
)
e R
n+2
:
e R}
P oposi ion
3
:
X

X
S
n
x
R
2
1--
.

S
n
x R+ x R
?
>
Rn+l
{0}
x
R
a) I
1I
:j+l_

>
Sn
.
is
a
2- la
subme sion
hen
X0(11J)
is a
ho izon
embeddin
g,
wi h
base-line

L,

such
ha

(XO(II,F))
1
(L)
_
0
.
b)
I
:Mn+1
.
-i
Rn+2
is a
ho izon
embedding
wi h
base-line
L
such
ha

-1
(L)
= 0
hen

11
:M
n+1
-
S
n
gi en
by
subme sion
.
(11
11
deno es
he
s anda d
no m
in
R

) .
P oo
:
a) I
111
is 2- la hen
(11,F)
is
áns e se
o
{
y} x
R
2
,
o
e e y
YES
n
.

I -
is
easy
o
check
.
ha
X0(11J)
is
also
ans e se
o
each

2-plane
con aining
L
and
he e o e
no a ine
angen
(n+1)-plane
o
X0(II,F)(M)
con ains
L
.
Tha is
o
say,
X0(11J)
is a
ho izon
embedding
.
b)
We
eplace
by
'
:M
n+1
=
.
Rn+1
{0}
x
R
.
Using
X1
1
and
X21
we
ob ain
a
map
M ->
S
n
x
R2
o
which
he
i s
componen
is
11
and he
second
one has he
equi edp ope y
o
2- la ness
.
We ema k
ha
11
has
no
c i ical
poin s
because
no
angen
(n+1) -
plane
o
(M)
con ains
L
.
11(x)
=
( 1(x),
..
., n+1(x))
/11
( 1(X),
.
.1, n+1(x))11
is a
2- la
n+2
Co olla y
2
:
I
11
:0
+1
-->
S
n
x
S1
.
P oo
:
Robe son
and
Chillingwo h
ha e
shown
E2]
3]
ha
i

:M
n+1
--*
Rn+2,
n
>
1,
is a
ho izon
imme sion
wi h
base-line

L,
such
ha

-1 (L) = 0
hen

M
is
di eomo phic
o
he
Klein
bo le o
S
n
x
S
1
.Howe e
he
Klein
bo le
canno
admi
a
2- la
subme sion
because
i
canno be embeddedin
3-space
.
S
nis a
2- la
subme sion
hen
M
is
di eomo phic
o
4
.
Embeddíngs
in eal
p ojec i en-space
RP
n
.
In
El]
i
was
shown
ha
i
11
:5
1
x S
1
--+
S
1
is

2- la
i is
possible
o
cons uc
an
embedding
S 1
x S1
-+
RP
3
,
ela ed
o
he
Hop
map
h
:S
3
-+
S2
.
In
his
sec ion
we
show
how
o
ob ain
an
embedding
M
m
-
.
RP
n+k

i
we
ha e
a

k- la
subme sion

1
:M
m
-
.
S
n
.

Le ,
us ake

F :~P
---u
Rk
,
associa ed
wi h
1I
l
necessa y,
we
eplace

F

by

F'
:¿
n
-+
Rk
such
ha

F'(M)
G
R+k
.
We ake
T
:S
n x
R
k
-+
RP
n
de ined
by T(x,y)
= Lx,y]
,

whe e
Lx,y]
is
he
1-dimen_
sional
subspace
o
Rn+k+l
de e minad
by
(x,y)
.
The
ank
o
Y'
is
n+k
and
i is
s aigh o wa d

o
check
ha

Y
Y
0(11,F)

is
an
embedding
.
Thanks
a e due
o
S ewa
A
.Robe son
o
sugges ing
he
p oo
o
p o-
posi ion
2,
much
simple
han
ou
p e ious
one
.
Re e ences
Robe son,S
.A
.
-
Pe mu a ions
associa ed
wi h
c i
.ical
poin s
on su aces
J
.London
Ma h
.Soc
.37
(1962),
329-337
.
L2]
-
Robe son,S
.A
.
-
The-dual
o a
heigh
unc ion,
J
.London
Ma h
.Soc
.
(2),
8
(1974),
187-192
.
13]
-
Robe son,S
.A,
and
Chillingwo h,
D .R
.J
.
-
Ho izon
maps
( o
appea )
.
E4]
-
Spanie ,
E .H
.
-
Algeb n
.a,¿c
Topozogy,

Ta a
McG aw-Hill,
New
Delhi
(1966)