Symmetry in Maps

1
Symmetry in Maps
2
Notation

• ?0 L
• ?1 R
• ?2 T
• T (1 2)(3 4)
• L (2 3)(1 4)
• R (2 3)(1 4)

R
L
T
3
Mon(M), Or(M)

• Mon(M) ltT,L,Rgt
• Example
• T (1 8)(2 3)(4 5)(6 7)
• L (1 4)(2 7)(3 6)(5 8)
• R (1 2)(3 4)(5 6)(7 8)
• Or(M) ltTR,RLgt
• ind IndMon(M)Or(M) 2.
• ind 1 ... nonorientable
• ind 2 ... orientable
• Mon(M) 4E (if M acts transitively on ?M)

2
3
1
4
8
5
6
7
B2 in the torus.
4
Morphisms of Maps

• f M ! N is a map morphism if f (g,z)
• g Mon(M) ! Mon(N)
• g - homomorphism
• g(T) T, g(L) L, g(R) R.
• z ?M ! ?N.
• z(Wx) g(W)(zx), x 2 ?M, W 2 Mon(M).
• g,z - both surjective.
• f - isomorphism, if both b,z bijective.

5
Theorem

• Theorem Morphisms between maps M and N are in
  one to one correspondence with covering
  projections between graphs Co(M) and Co(N).

6
Aut(M)

• Group of map automorphisms.
• Aut(M) Mon(M)
• Aut(M) 4E(M) Mon(M).
• In our example
• Aut(M) 8 Mon(M).
• Aut(M) Mon(M).

7
Dual Revisited

• By interchanging the role of T and L we obtain
  the dual.
• The map on the left is self-dual. This means that
  M is isomorphic to Du(M) by f.
• f(1) 6, f(2) 5, f(3) 8, f(4) 7, f(5)
  2, f(6) 1, f(7) 4, f(8) 3.
• f2 id.

2
3
1
4
8
5
6
7
B2 in the torus and its dual.
8
Aut(M) 4E(M)

• Theorem. Let x 2 ?M. Each ? 2 Aut(M) is
  determined by y 2 ?M such that y ?(x).
• CorollaryAut(M) ?M 4E(M).
• A map M with Aut(M) 4E(M) is called
  reflexive.

9
Edge-transitive Map

• A map M is edge-transitive if Aut(M) acts
  transitively on E(M).
• Note The 1-skeleton G(M) of an edge-transitive
  map is an edge-transitive graph. The converse is
  not true in general.

10
Homework

• H1. Determine the Petrie dual of our example.

11
Non-degenerate edge-transitive maps

• A map is non-degenerate if and only if the
  minimal valence of graph, dual graph and petrie
  graph is at least 3.

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The Petrie dual

• Each map is defined by three involutions on flags
  (t0,t1,t2). Now add the product t3t0t2, that is
  another fixedpoint free involution. This can be
  viewed as an rank 4 incidence geometry
  (t0,t1,t2,t3).
• Orbits for ltt1,t2gt form the vertex set V.
• Orbits for ltt0,t2gt form the edge set E.
• Orbits for ltt0,t1gt form the face set F.
• Orbits for ltt1,t3gt form the Petrie walks P.

Du
V
F
E
Op
Pe
P
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The Petrie hexagon
M

• M (t0,t1,t2,t3)
• Du(M) (t2,t1,t0,t3)
• Pe(M) (t0,t1,t3,t2)
• Du(Pe(M)) (t3,t1,t0,t2)
• Pe(Du(M)) (t2,t1,t3,t0)
• Pe(Du(Pe(M))) Du(Pe(Du(M))) (t0,t1,t3,t2)

Du(M)
Pe(M)
Du(Pe(M))
Pe(Du(M))
Pe(Du(Pe(M))) Du(Pe(Du(M)))
14
Group Or(M) revisited

• Or(M) contains all even words. It acts on ?. If
  the action on Or(M) has two orbits, then we may
  partition the set of flags into two subsetes ?
  and ?-.
• M orientable iff Or(M) has TWO orbits.

15
Local Automorphisms

• Rooted maps. (Maps rooted in a flag!!!)
• Local automorphisms (around the edge)
• There are 14 possible types.

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Local situation - Notation

• i - (identity flag)
• e - edge
• x1 - close vertex
• x2 - far vertex
• f1 - close face
• f2 - far face

f1
i
e
x1
x2
f2
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Local automorphisms - Notation
?2 i

• i - (identity flag)
• Involutions ? i, ? i, ? i
• Rotations ?x1 i, ?f1 i, ?1 i
• Involutions q1 i, q2 i, q3 i, q4 i,
• Rotations ?x2 i, ?f2 i, ?2 i
• Exercise Draw the missing three rotations in
  the Figure on the left.

?f1 i
f1
?1 i
li
i
sx1i
e
x1
x2
fi
?i
f2
?1 i
?3 i
?4 i
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Formal Definitions

1. id
2. ?x1 TR
3. ?x2 LRTL
4. ?f1 RL
5. ?f2 LTRT
6. g1 RTL
7. g2 LRT
8. q1 R
9. q2 LRL
10. q3 TRT
11. q4 LTRTL
12. t T
13. l L
14. f TR

• Each of the fourteen elements of Mon(M) on the
  left can be expressed as a word in T,R,L.
• We are sure that id 2 Aut(M). However, other
  elements may or may not belong to Aut(M).

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Type of edge-transitive map

• 1
• 2
• 2P
• 2ex
• 2Pex
• 3
• 4
• 4
• 4P
• 5
• 5
• 5P

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Map symbol of edge-transitive map

• (abc) 1
• (aa'bc) 2
• (abb'c) 2
• (abcc') 2P
• (abc) 2ex
• (abc) 2ex
• (abc) 2Pex
• (aa'bb'cc') 3
• (aa'bc) 4
• (abb'c) 4
• (abcc') 4P
• 5
• 5
• 5P

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Facts

• Du(1)

22
Small example

• On the left we see a map on the Klein bottle.

23
Homework

• H1. Prove that in a reflexive rooted map the
  group lt?,?,?1gt acts transitively on the flags.

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Coverings of combinatorial maps

• Each morphism of maps is a covering projection.

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Lifting automorphisms
f 2 Aut X
X
X
p
p
f 2 Aut X
X
X
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CT(X)

• CT(X) it the group of covering transformations.

27
Regular Covers

• Covering is regular if and only if CT(X) acts
  regularly on the fibers p-1(x).

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Voltages for maps

• Combinatorial map Co(M) of M with a given edge
  color T,L,R.
• Each edge lifts to an edge of the same color.
• Voltage ? E(Tr(M)) ? ?.
• Instead of edge we assign voltages to the colored
  edges.
• ?(Te) ?(e)-1.
• ?(Le) ?(e)-1.
• ?(Re) ?(e)-1.
• ?(e)?(Le) ?(e)?(Te). edges to edges...

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Voltages for Maps

• ? (?,?,?).
• For each ?, ?, ? we get a map.

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Example 1
31
Theorem

• Let M be a map, G a group and ?Co(M) ! G a
  voltage assignment. Let M be the derived map.
• Let f 2 Aut(M)