Symmetry in Maps

1. Symmetry in Maps

2. 3 2 4 1 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) = <T,L,R> • 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) = <TR,RL> • ind = Ind[Mon(M):Or(M)] · 2. • ind = 1 ... nonorientable • ind = 2 ... orientable • |Mon(M)| ¸ 4|E| (if M acts transitively on M) 3 2 4 1 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. 3 2 4 1 8 5 6 7 B2 in the torus. Aut(M) • Group of map automorphisms. • Aut(M) · Mon(M) • |Aut(M)| · 4|E(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. 3 2 4 1 8 5 6 7 B2 in the torus and its dual.

8. |Aut(M)| · 4|E(M)| • Theorem. Let x 2M. Each 2 Aut(M) is determined by y 2M such that y = (x). • Corollary:|Aut(M)| · |M| = 4|E(M)|. • A map M with |Aut(M)| = 4|E(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.

12. The Petrie dual • Each map is defined by three involutions on flags (t0,t1,t2). Now add the product t3=t0t2, that is another fixedpoint free involution. This can be viewed as an rank 4 incidence geometry: (t0,t1,t2,t3). • Orbits for <t1,t2> form the vertex set V. • Orbits for <t0,t2> form the edge set E. • Orbits for <t0,t1> form the face set F. • Orbits for <t1,t3> form the Petrie walks P. Du V F E Op Pe P

13. 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.

16. 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

17. 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

18. id = x1 = TR x2 = LRTL f1 = RL f2 = LTRT g1 = RTL g2 = LRT q1 = R q2 = LRL q3 = TRT q4 = LTRTL t = T l = L 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). Formal Definitions

19. Type of edge-transitive map • 1 • 2* • 2P • 2ex • 2Pex • 3 • 4 • 4* • 4P • 5 • 5* • 5P

20. Map symbol of edge-transitive map • (a:b:c) 1 • (a:a':b:c) 2 • (a:b:b':c) 2* • (a:b:c:c') 2P • (a:b:c) 2ex • (a:b:c) 2*ex • (a:b:c) 2Pex • (a:a':b:b':c:c') 3 • (a:a':b:c) 4 • (a:b:b':c) 4* • (a:b:c:c') 4P • 5 • 5* • 5P

21. 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 <,,1> acts transitively on the flags.

24. Coverings of combinatorial maps • Each morphism of maps is a covering projection.

25. Lifting automorphisms f~2 Aut X~ X~ X~ p p f 2 Aut X X X

26. 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).

28. 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...]

29. Voltages for Maps •  = (,,). • For each , ,  we get a map.

30. 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)