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https://doi.org/10.1007/s00283-022-10214-w
359
Trisecting theNine‑Vertex Complex
Projective Plane RichardEvanSchwartz
A
k-simplex is a k-dimensional convex polytope
with
k+1
vertices. For
k=0, 1, 2, 3
, a
k-simplex is usually called respectively a
vertex, edge, triangle, tetrahedron. When k
is not important, a k-simplex is just called a
simplex.
A simplicial complex is a nite collection
C
of simplices,
all in an ambient Euclidean space, such that
• If
S∈C
and
S′
is a subsimplex of S, then
S�∈C
.
• If
S,T∈C
, then
S∩T
is either empty or in
C
.
Informally, the simplices in a simplicial complex t together
cleanly, without crashing through each other. The support
|C|
of
C
is the union of all the simplices in
C
. Often we blur
the distinction between
C
and
|C|
and think of a simplicial
complex as a union of simplices.
A simplicial complex may be described with no mention
of the ambient space containing it, but there is always the
understanding that in principle, one can nd an isomor-
phic complex in some Euclidean space. To give a pertinent
example, let
RP2
6
be the quotient of the regular icosahe-
dron by the antipodal map. This simplicial complex has 6
vertices, 15 edges, and 10 faces. One can reconstruct
RP2
6
in
ℝ5
by xing some 5-simplex
Σ⊂ℝ5
, the convex hull of
vertices
v1,…,v6
, then mapping vertex k of
RP2
6
to
vk
and
extending linearly.
The left side of Figure1 shows another incarnation of
RP2
6
. In this picture, the outer edges of the hexagon are sup-
posed to be identied according to the labels. The complex
RP2
6
is called a six-vertex triangulation of the real projective
plane
RP2
because its support is homeomorphic to
RP2
.
This triangulation has the fewest number of vertices among
triangulations of
RP2
, so it is called a minimal triangulation
of
RP2
. It is, in fact, the unique minimal triangulation of
RP2
. (Smaller examples such as the quotient of the regu-
lar octahedron by the antipodal map fail to be simplicial
complexes.)
Here are some other examples related to minimal
triangulations.
• The boundary of a tetrahedron is the unique four-vertex
minimal triangulation of the 2-sphere. More generally, the
boundary of a
(k+1)
-simplex is the unique minimal trian-
gulation of the k-sphere.
• If you identify the opposite sides of the big hexagon on
the right side of Figure1, you get the unique minimal tri-
angulation
T2
7
of the 2-torus. It has 14 triangles, 21 edges,
and 7 vertices.
• In 1980, Wofgang Kühnel discovered
CP2
9
, the unique
nine-vertex minimal triangulation of the complex pro-
jective plane
CP2
. This triangulation has 36 4-simplices
and a symmetry group of order 54.
• In 1992, Ulrich Brehm and Wolfgang Kühnel [4] dened
HP2
15
(and two variants), a 15-vertex simplicial complex
with 490 8-simplices. In 2019, Denis Gorodkov [6] proved
that
HP15
and the variants are PL homeomorphic to the
quaternionic projective plane
HP2
.
• So far, it an open question whether there is a 27-vertex
triangulation of
OP2
, the octonionic (aka Cayley) projec-
tive plane.
• The minimal triangulations of
RP3
and
RP4
have respec-
tively 11 and 16 vertices. See [5].
• In 2021, Karim Adiprasito, Sergey Avvakumov, and
Roman Karasev [1] proved that real projective space
can be triangulated using a subexponential number of
simplices.
The survey article [5] by Basudeb Datta has a wealth of
information about minimal triangulations up to the year
2007 and a large number of references.
The subject of this paper is
CP9
2
. In [7], Wolfgang Kühnel
and Thomas F. Bancho establish many interesting proper-
ties of
CP2
9
and give a rather intricate proof that
CP2
9
really is
homeomorphic to
CP2
. Since [7], there has been a lot of work
done trying to understand
CP2
9
from various points of view.
In particular, there are a number of proofs that
CP2
9
≅CP
2
,
and also a number of proofs that
CP2
9
is the only minimal
triangulation of
CP2
. See the article by Bernard Morin and
Figure1.
RP2
6
(left) and
T2
7
(right).
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