Symmetry in Maps

Notation 0 = L 1 = R 2 = T T = (1 2)(3 4) L = (2 3)(1 4)

Mon(M), Or(M) Mon(M) = <T,L,R> Example: Or(M) = <TR,RL> 3 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 5 8 6 7 B2 in the torus.

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.

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

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). 1 2 3 4 5 6 7 8 B2 in the torus.

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 5 8 6 7 B2 in the torus and its dual.

|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.

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.]

Homework H1. Determine the Petrie dual of our example.

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.

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

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)))

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.

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

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

Local automorphisms - Notation f1 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 1 i i li sx1i e x1 x2 i fi f2 1 i 3 i 4 i

Formal Definitions 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).

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

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

Facts Du(1) =

Small example On the left we see a map on the Klein bottle.

Homework H1. Prove that in a reflexive rooted map the group <,,1> acts transitively on the flags.

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

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

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

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

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...]

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

Example 1

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)