Journal of Geometry
Vol. ii/i 1978
Birkh~user Verlag Basel
THE TRANSPOSITION OF LOCALLY COMPACT,
CONNECTED TRANSLATION PLANES
Thomas Buchanan
and
Hermann H~hl
This note deals with the transposition of translation
planes in the topological context.
We show that a topological congruence
C of the real vecto~ space R 2n has
the property that every hyperplane of ~zn
contains a
component of C. This makes it possible to define the
transpose
P~ of the topological translation plane
P
associated with
C; it is proved that the translation
plane
pT is topological also.
The relationship between
collineation groups and the relationship between coordinatizing quasifields of P and
pT are also discussed.
w
Before we formulate our main results, we recall some facts
and definitions.
The affine point set of a locally compact,
topological translation plane
P
connected,
can be identified topo-
logically with a finite dimensional real vector space
(in its usual topology)
through the origin of
in such a manner that the lines
V
are linear subspaces ([3, p.12-13]).
These lines form a congruence of
plane
P
V;
and the translation
is the translation plane associated with this
congruence in the sense of Andr@ [I].
real finite dimensional vector space
Any congruence of a
V
which can be ob-
tained in this manner will be called a topological
gruence
sions of
By
V
of
V.
V
V*
are
con-
It is known that the only possible dimen2, 4, 8
and
16
([3, p.47 Satz I]).
we denote the dual space of
84
V
consisting of
2
BUCHANAN
all linear forms
space
C
of
V
{~ 6 V* / ~(C)
V
, ~;
by the polar of a linear sub-
we mean as usual the annihilator
= O}
Using results
of
C
in
C
[3] we shall prove
be a topological congruence of a
real finite dimensional vector space
(a)
C
the set
Remark:
C,
Then
C,
V
and
of the polars of the
C ~ : {C ~ / C 6 C}
components of
V ~,
V.
has the property that any hyperplane Of
contains a component of
(b)
C~ =
V*.
of Breuning
Let
THEOREM.
and H~HL
which by
(a)
i8 a congruence of
is a topological congruence.
The topological
be omitted,
hypothesis
since by [2, Teorema
sarily nontopological)
have the hyperplane
Let
COROLLARY.
2.2]
congruences
property
P
of our theorem
of
of part
there exist
V
cannot
(neces-
which do not
(a).
be a locally compact, connected,
topological translation plane, and let
C
be the
associated topological congruence of the underlying
real vector space
pT
V.
Then the translation plane
associated with the corresponding polar con-
gruence
Cb
is also a locally compact, connected,
topological translation plane.
The plane
is called the transpose I) of
pT
Proof of the Theorem.
(a) Identify V with
~2n
and give
C
P.
the finest
topo-
logy such that the map
(I)
R2n\{o}
, C: x ; ~ C
,
x
which sends each nonzero vector
x
nent
is continuous.
C
of
x
of [5, p.14]
C
containing
x,
C
is homeomorphic
to the unique
to the n-sphere
compoBy Satz 2
S n.
I) The authors are indebted to Dr. Rainer L6wen for
calling their attention to this elegant description
of the transpose using polars.
85
BUCHANAN
Moreover,
if we consider
manifold
Gn(N2n)
C
of
C
Let
H
component
of
C.
component
of
C
of dimension
that
Then obviously
n-l.
induced
from
but fixed hyperplane
we assume
would
N2n
I and p.17]).
be an arbitrary
indirectly,
of the Grassmann
subspaces of
is the topology
([5, p.50 Satz
Arguing
5
as a subset
of n-dimensional
then the topology
Gn(~2n)
and H~HL
H
n Z 2.
intersect
H
in
~2n.
does not contain
Moreover,
a
every
in a linear subspace
Thus
CIH = {C nH / C 6 C}
would be a s u b s e t
o f t h e Grassmann m a n i f o l d
looking
at the Grassmann
Stiefel
manifolds
(2)
manifolds
C ---+ Gn_I(H):
C I
Thus
would be homeomorphic
to
the canonical
over
vector
total
:
{(CAll,x)
vector bun-
for example
in
yn-l(H)
and by removing
the zero
obtain
CIH
a locally
with fiber
to
(CxD H,x),
(2)
to
trivial
Hn-l\{o},
n Z 2,
This
E
would be homeH\{O}
~ E:
using maps
(I)
We now note an inconsist-
the following
part
of the exact homo-
of this fiber bundle:
--+ ~n_l(S n) ~
nn_2(~2n-1\{0})
space
by the map
can be expressed
continuity.
x 6 C},
given by the projection
The total
= ~2n-l\{O}
which
to check
topy sequence
E - , CIH
factor.
H\{0}
/
6 (CIH)• (H-{O})
projection
ency when we examine
zero.
Gn_I(H)
By restricting
we would
over the n-sphere
onto the first
omorphic
since
of
space
and bundle
and
as defined
Gn_I(H)
from each fiber,
E
x I,
of
as a subspace
(n-l)-dimensional
Gn_l(H)
[6, P.59 ff.].
CIH
fiber bundle
~ CA H
CIH
Consider
the subspace
of
S n.
xn-l(H)
Milnor-Stasheff
By
spaces
it is easy to see that the map
would be continuous.
dle
Gn_I(H).
as quotient
nn_2(~n-l\{o } ) --+ nn_ 2 (~2n-l~{O})
the homotopy
groups
are zero, whereas
contradiction
proves
86
(or sets)
nn_l(S n)
Un_2(~n-l\{0})
part
__+ "
and
is not
(a) of the theorem.
4
BUCHANAN
(b)
By [6, P.57 Lemma
into the Grassmann
V*,
to
Sn
C~
complement
of
is nothing
Therefore
C
since
spaces
V
more than the
C~
is homeomorphic
the proof of part
the following
LEMMA.
Let
C
ar subspaces
The "only
lemma, which
as components.
C
if" part of this
(a)
of the proof;
E
=
{(C,x)
to
E
C
having
/
[3, p.27 Satz,
p.28 Korollar
been used in
from
vector bundle
space
x 9 C}
(in the terminology
E ---+ ~2n:
Gn(~2n).
For the "if"
the total
~ C: (C,x) I
tive Gauss map
line-
is topological
n-dimensional
having
in [3].
immediately
I and p.17].
9 C •
and the projection
admits
C
~2n
lemma has already
we look at the canonical
restricted
it suffices
as a subset of
it follows
[3, p.14 Satz 2, p.30 Satz
n(~2n)
of
Then
is compact
(b)
is implicit
b e a congruence
if and only if
bundle
from
too.
to apply
part,
C~
is continuous,
of the vector
C
C•
Thus to complete
part
C ~-+
Gn(V*)
identification
the polar
orthogonal
5.1] the map
manifold
after appropriate
and
and HIHL
, C.
This vector
of [3, p.22])
(C,x)
i ~ x.
the effec-
We can then apply
and p.34 Satz 1].
w
In this section
transposition
plane
we make
a few remarks
of a locally
compact,
as far as the collineation
quasifields
Recall
ly compact,
groups
of
translation
and coordinatizing
are concerned.
that
a quasifield
connected
compact,
connected,
Salzmann
[7, w
lation plane.
which
translation
topological
conversely,
such a quasifield
topological
about the effects
connected
The kernel
~, C
K
P
quasifield
any plane
is a locally
field to
coordinatizes
plane
compact,
of
P
in the sense
coordinatized
connected
87
of
by
trans-
is isomorphic
or the quaternion
([3, p.8 Satz 3]).
a local-
is a locally
field
as a
B U C H A N A N and H ~ H L
5
We first claim that the transpose
c o n s t r u c t e d equally well using
K
pT
of
instead of
P
can be
R
as the
base field.
More generally
let
P
be a (not necessarily
topolo-
gical) t r a n s l a t i o n plane which is finite dimensional over
its kernel
K.
By
[I] the affine point set~ V
of
P
may
be considered as a one-sided vector space over
K
in such
a m a n n e r that the components of the congruence
C
associ-
ated with
P
P
are
is locally
K - l i n e a r subspaces.
compact and connected,
(In the case where
the
R - l i n e a r struc-
ture w h i c h we have been c o n s i d e r i n g up to now is, of
course,
of
V
simply given by r e s t r i c t i o n to the closed subfield
K.)
as a
Given any subfield
of
K,
we can regard
K ' - v e c t o r space and form its dual
HomK,(V,K')
VK , =
over
K'
K'
For
C e C
CK, :
K'-polar
consider the
0
I
{~ 6 VK,
r
= O}
family
and the
O
0
CK, :
{CK,
/
PROPOSITION
1.
F
cec}.
Suppose
K'
a central subfield of
nite dimension
ists an
C~,
K'
such that
over all these fields.
F-linear isomorphism
Co
F
onto
is a subfield of
for every
one of the families
C F0
V
and
K
has
is
ex-
which maps
~ VF
In particular,
o
CK,
and
fi-
Then there
,
C 6 C.
V
if
a congruence,
the other i8 also.
Proof:
If
6: K'
A: VK,
~ F
is any nonzero
K'.
for every
C
The map
E
V K, are equal,
equal,
a
let
' VF: ~ ~ ' ~
be the induced map, which is
tral in
F - l i n e a r form,
C.
A
F-linear,
is injective,
Since the
and the
Let
P
and
F-dimensions
F-dimensions of
must be an isomorphism,
COROLLARY.
since
and
be a locally
88
CF
~
F
A(C
of
VF
compactj
o
CF
and
CoO
K'
= CF .
and
A(C~,)
is cen,) ~
are
connected
6
BUCHANAN
translation plane,
K
ence associated with
congruence,
and H~HL
its kernel and
C the congruo
P. Then
C K is a topological
and the translation plane associated with
C Ko is topologically isomorphic to the transpose
described in the Corollary on page ~.
This
1
follows
with
from the Theorem on page 2 using Proposition
F = 8
Maduram
and
K' = K.
[5] has described
some collineation
fields
groups
locally
results
compact,
topological
analogues
indications
planes
between
coordinatizing
translation
carry over to
planes.
We give
of three of his propositions,
interest
in this
quasi-
and their transposes.
as well as his proofs
connected
are of p a r t i c u l a r
the relationships
and between
of finite translation
His general
pT
context,
which
together with
of proofs using our notation.
PROPOSITION
2 (cf.
cally compact,
[5, Prop.
3]).
Let
P
be a lo-
connected translation plane,
let
F0
denote the isotropy subgroup of the affine collineation group of
let
~0
P
with respect to the origin, and
denote the subgroup of continuous
ation8 in
F0;
9r *0
let
and
colline-
be the analogously
E0
*
defined groups of collineations of the transpose
Then
F0
and
P0
pT.
are isomorphic--viewed as trans-
formation groups of the congruences associated with
P
and
maps
pT
E0
respectively--via
onto
lation group of
tation of the
pT
of
companion
automorphism,
tion
of
(3)
V~
AV(~)
on the trans-
~0
on the trans-
P.
V
([1, p. IZ8 Satz 19]). For
Av
gO
is the contragredient represen-
recall that
linear automorphisms
C
9 . The action of
E0
R-linear action of
lation group of
For the proof,
an isomorphism which
r0
consists
which respect
A 6 r0
define a
with
(~ 6 V~).
89
K-semi-
the congruence
a 6 Aut K
K-semilinear
by
= ao~oA -I
of all
its
transforma-
BUCHANAN and HEHL
7
A s t r a i g h t f o r w a r d calculation shows that
Av
respects the
o
congruence
C K , and thus by the corollary on page 5 may
@,
be considered as an element
of
F O.
One easily checks
@
that
A e--+ AV:
F0
, F0
and the b i j e c t i o n
is a group isomorphism;
o. C : ~ C K
o
.+ C K.
C
this
t o g e t h e r constitute
a t r a n s f o r m a t i o n group i s o m o r p h i s m
(4)
(to,C)
~
(r *,C~):
o
(A,C)
Since the companion a u t o m o r p h i s m of
continuity of
A, a
and
Av
8: K
---* R
its real part
6(z)
: Re z
@
and identify
Av
E0
is again
and
a,
EO,
the
take the
w h i c h associates to each
(K
is either
~, C
z 6 K
or
~),
@
VK
and
V~
by the induced i s o m o r p h i s m
as d e f i n e d in the proof of Proposition
prete formula
(3)
~oa = 6
a
when
0
(A ,C K) .
are equivalent.
To compare the actions of
R - l i n e a r form
v
~
in
V~:
I.
A
We shall inter-
M a k i n g use of the fact that
is a continuous a u t o m o r p h i s m of
K,
we
have
AoAMoA-I(e)
@
= ~ o a o ( A - l e ) o A -I = ~oA -I
( A E n0, ~ 6 V ~ ) ,
w h i c h is the expression d e f i n i n g the contragredient
repre-
sentation.
P R O P O S I T I O N 3 (cf.
[5, Prop.
be a locally compact,
finite dimensional
the translation
division
(respectively,
coordinatizes
same dimension
compact,
as
Q,
of
Remark:
Q
and let
by
Q
(or a real
Q.
P
be
Then
connected quasifield
a real division
the transpose
and the left nucleus
morphic
algebra),
plane coordinatized
there i8 a locally
Let
4 and 5(3)]).
connected quasifield
algebra)
pT
Q',
which
and thus has the
such that the middle nucleus
of
Q'
to the left nucleus
are topologically
iso-
and the middle nucleus
respectively.
Note that Dembowski,
Andr~ and Maduram consider
quasifields with the distributive
law
(a + b)c = ac + bc,
whereas the d i s t r i b u t i v e
law used by B r e u n i n g and Salzmann
is
Proposition
c(a + b) = ca + cb.
90
3
is formulated ac-
8
BUCHANAN
and H~HL
cording to the latter convention;
in particular,
the left
nucleus here plays the role of the right nucleus in [4, p.
134]
and [5].
For the proof of the proposition,
let
denote the first and second coordinate
dinatization
points
of
P
over
at infinity
Pl
Q,
C1
and
C2
axes in the coor-
and call their respective
and
P2"
Take
Q'
to be a lo-
cally compact,
connected quasifield which coordinatizes
with respect to the K-polars
(Ci) Ko 6 CKo (i : 1,2)
pT
as first and second coordinate
axes.
To simplify notao.
C i, denote their respective
tion we write these axes as
o
Pi
(i = 1,2).
points at infinity by
By straightforward
formation
stricts
calculation
group isomorphism
(F0,C)
to an isomorphism between
we see that the transo
~ (F0,C K)
the group
of
(4) re-
r(Pi,C ~)
of
V
central collineations of
and the group
r*( pj,
o cO,
i)
with center
pjo
P
with center
C?i
and axis
(i,j 6 {1,2}).
groups of central collineations
Satz 22]); hence the groups
and the isomorphisms
proof proceeds
(cf.
terchange
Q'
are
K-linear
carry the obvious
above are clearly
by using the well-known
these groups to substructures
fields
[4, p.134,
Pi
and axis
of central collineations
of
C.
P TJ
These
([I, p. 180
topology,
continuous.
The
relationships
of the coordinatizing
of
quasi-
3.1.30 ff.]): The left-middle
in the assertion regarding the nuclei of
inQ
and
follows from the isomorphisms
F(PI,C2) ~ F*(P2,Cl~o)
o o
F(P2,C I) m F*(Pl,C 2) and the relationships of these
and
strain groups to nuclei.
when
Q
tive on
Moreover,
is, since the shears group
C ~{C 2}
Q'
is a semifield
s
is transi-
if and only if the corresponding
group
r*(
o
o
o
P2,C2)
is transitive on C Ko \{C2}.
Finally, a locally compact, connected semifield is a real finite dimen-
sional
(not necessarily
because by the arguments
associative)
division
algebra,
of [5, p.8 Satz 3] it contains
in its center.
The contribution of the second author to this paper
is part of his work on a program sponsored by the Deutsche
Forschungsgemeinschaft,
and he would like to thank the DFG
91
BUCHANAN and H%HL
9
for their support.
REFERENCES
1. Andre, J.: Ober nicht-desarguessche Ebenen mit transitiver Translationsgruppe.
Math. Z. 60 (1954), 156-186.
2. Bernardi, M.: Esistenza di fibrazioni in uno spazio
proiettivo infinito.
!st. Lombardo Acad.Sci. Lett. Rend.
A 107 (1973), 528-542.
3. Breuning, P.: Translationsebenen und VektorraumbHndel.
Mitt.Math.Sem. Giessen 86 (1970), 1-50.
4. Dembowski, P.: Finite Geometries.
New York, Springer-Verlag 1968.
Berlin-Heidelberg-
5. Maduram, D.M.: Transposed translation planes.
Proc.
Amer. Math. Soc. 53 (1975), 265-270.
6. Milnor, J., Stasheff, J.D.: Characteristic Classes.
Princeton, Princeton University Press 1974.
7. Salzmann, H.: Topological planes.
(1967), 1-60.
Advances in Math. 2
Thomas Buchanan, Hermann H~hl
Mathematisches Institut
der Universit~t T~bingen
Auf der Morgenstelle 10
D-7400 TGbingen I
Federal Republic of Germany
(Eingegangen am 3. Februar 1977)
92