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On automorphism groups of maps, surfaces and Smarandache geometries (1).

Abstract A combinatorial map is a connected topological graph cellularly embedded in a surface. This report concentrates on the automorphism group of a map, which is related to the automorphism groups of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the automorphism groups of maps, Klein surfaces and Smarandache manifolds and the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An classification for the closed s-manifolds by maps is found. Open problems related the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this report for the further applications of the combinatorial maps to the classical mathematics.

Keywords Automorphism group; Surface; Map; Smarandache geometries; Map geometries; Classification.

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Part I. Terminology and Notations

[section] 1.1 Klein Surfaces

A Klein surface is a Hausdorff, connected, topological space S with a family [summation] = {([U.sub.i], [[empty set].sub.i])| i [member of]I} such that the chart {[U.sub.i]|i [member of] I} is an open covering of S, each map [[empty set].sub.i] : [U.sub.i] [right arrow] [A.sub.i] is a homeomorphism onto an open subset [A.sub.i] of C or [C.sup.+] = {z [member of] C : Imz [greater than or equal to] 0} and the transition functions

[[empty set].sub.ij] = [[empty set].sub.i] [[empty set].sup.-.sub.j] : [[empty set].sub.j] ([U.sub.i][intersection][U.sub.j]).

are dianalytic, where a mapping f : A [right arrow] C is said dianalytic if [partial derivative] f / [partial derivative] z = 0 (Cauchy-Riemann equation) or [partial derivative] f / [partial derivative] z = 0.

[section] 1.2 {Riemann Surfaces} [subset] {Klein surfaces}

[section] 1.3 Embedding and Combinatorial Maps Embedding of a graph:

For any connected graph [GAMMA] = (V ([GAMMA]), E([GAMMA])) and a surface S, an embedding of the graph [GAMMA] in the surface S is geometrical defined to be a continuous 1 - 1 mapping [tau] : [GAMMA] [right arrow] S. The image [tau]([GAMMA]) is contained in the 1-skeleton of a triangulation of the surface S. If each component in S - [tau] ([GAMMA]) homeomorphic to an open disk, then the embedding is an embedding.

Map:

A combinatorial map is a connected topological graph cellularly embedded in a surface.

The Algebraic Definition of Maps:

A combinatorial map M = (X([alpha],[beta], P) is defined to be a basic permutation P, i.e, for any x [member of] [X.sub.[alpha], [beta]], no integer k exists such that [P.sup.k.sub.x] = [alpha]x, acting on [X.sub.[alpha],[beta]], the disjoint union of quadricells K x of x [member of] X (the base set), where K = {1, [alpha], [beta], [alpha][beta]} is the Klein group, satisfying the following two conditions:

(Ci) [alpha]P = [P.sup.-1][alpha];

(Cii) the group [[PSI].sub.J] =< [alpha], [beta], P > is transitive on [X.sub.[alpha], [beta]].

[section] 1.4 Orientation

If the group [[PSI].sub.I] =< [alpha][beta], P > is transitive on [X.sub.[alpha], [beta]], then M is non-orientable. Otherwise, orientable.

[section] 1.5 An Example of Maps [K.sub.4] on the torus.

[FIGURE 1 OMITTED]

([X.sub.[alpha]],[beta], P):

([X.sub.[alpha]],[beta] = {x, y, z, u, v, w, [alpha]x, [alpha]y, [alpha]z, [alpha]u, [alpha]v, [alpha]w, [beta]x, [beta]y, [beta]z, [beta]u, [beta]v, [beta]w, [alpha][beta]x, [alpha][beta]y, [alpha][beta]z, [alpha][beta]u, [alpha][beta]v, [alpha][beta]w}

P = (x, y, z) ([alpha][beta]x, u, w) ([alpha][beta]z, [alpha][beta]u, v) x ([alpha][beta]y, [alpha][beta]v, [alpha][beta]w) ([alpha]x, [alpha]z, [alpha]y) ([beta]x, [alpha]w, [alpha]u) x ([beta]z, [alpha]v, [beta]u) ([beta]y, [beta]w, [beta]v)

Vertices:

[v.sub.1] = {(x, y, z), ([alpha])x, [alpha]z, [alpha]y)}

[v.sub.2] = {([alpha][beta]x, u, w), ([beta]x, [alpha]w, [alpha]u)}

[v.sub.3] = {([alpha][beta]z, [alpha][beta]u, v), ([beta]z, [alpha]v, [beta]u)}

[v.sub.4] = {([alpha][beta]y, [alpha][beta]v, [alpha][beta]w), ([beta]y, [beta]w, [beta]v)}

Edges:

{e, [alpha]e, [beta]e, [alpha][beta]e}, e [member of] {x, y, z, u, v ,w}

Faces:

[f.sub.1] = {(x, u, v, [alpha][beta]w, [alpha][beta]x, y, [alpha][beta]v, [alpha][beta]z), ([beta]x, [alpha]z, [alpha]v, [beta]y, [alpha]x, [alpha]w, [beta]v, [beta]u)}

[f.sub.2] = {(z, [alpha][beta]u, w, [alpha][beta]y), ([beta]z, [alpha]y, [beta]w, [alpha]u)}

[section] 1.6 Isomorphism of Maps

Two maps [M.sub.1 = ([X.sup.1.sub.[alpha],[beta]], [P.sub.1]) and [M.sub.2] = ([X.sup.2.sub.[alpha],[beta]], [P.sub.2]) are said to be isomorphic if there exists a bijection [xi]

[xi]: [X.sup.1.sub.[alpha],[beta]] [right arrow] [X.sup.2.sub.[alpha],[beta]]

such that for [for all]x [member of] [X.sup.1.sub.[alpha],[beta]],

[xi][alpha](x) = [alpha][xi](x), [xi][beta](x) = [beta][xi](x), [xi][P.sub.1](x) = [P.sub.2][xi](x).

[section] 1.7 Equivalence

Two maps [M.sub.1], [M.sub.2] underlying graph [GAMMA] are equivalent if there exists an isomorphism [zeta] between them induced by an element [xi], [xi] [member of] Aut[GAMMA]. Call [zeta] an equivalence between [M.sub.1], [M.sub.2]. Certainly, on an orientable surface, an equivalence preserve the orientation on this surface.

Theorem 1.1. Let M = ([X.sub.[alpha],[beta]], P) be a map with an underlying graph [GAMMA], [for all]g [member of] Aut[GAMMA]. Then the extend action of g on [X.sub.[alpha], [beta]], P] with X = E([GAMMA]) is an automorphism of map M iff [for all]v [member of] V (M), g preserves the cyclic order of v.

[section] 1.8 Covering of Maps

For two maps [??] = ([??], [??]) and M = ([X.sub.[alpha],[beta]], P), call the map [??] covering the map M if there is a mapping [pi] : [??] [right arrow] [X.sub.[alpha],[beta]] such that [for all]x [member of] [??],

[alpha][pi](x) = [pi][alpha](x), [beta][pi](x) = [pi][beta](x), [pi][??](x) = P [pi](x).

Theorem 1.2. Let [pi] : [??] [right arrow] [X.sub.[alpha],[beta]] be a covering mapping. Then [pi] is an isomorphism iff [pi] is an 1 - 1 mapping.

[section] 1.9 Voltage Map

Let M = ([X.sub.[alpha],[beta]], P) be a map and G a finite group. Call a pair (M, [??]) a voltage map with group G if [??] : [X.sub.[alpha],[beta]] [right arrow] G, satisfying the following condition:

(i) [for all] x [member of] [X.sub.[alpha],[beta]], [??]([alpha]x) = [??](x), [??]([alpha][beta]x) = [??]([beta]x) = [[??].sup.-1](x);

(ii) [for all] F = (x, y,..., z)([beta]z,..., [beta]y, [beta]x) [member of] F(M), the face set of M, [??](F) = [??](x) [??](y) ... [??](z) and < [??](F)|F [member of] F(u), u [member of] V (M) >= G, where, F(u) denotes all the faces incident with the vertex u.

[section] 1.10 Lifting of a Voltage Map

For a voltage map (M, [??]) with group G, the map [M.sup.[??]] = ([X.sup.[??].sub.[alpha],[beta]], [P.sup.[??]]) is called its lifting map.

Theorem 1.3. An finite group G is a fixed-free automorphism group of a map M = ([X.sub.[alpha],[beta]], P) on V (M) iff there is a voltage map (M/G, G) with an assignment [??] : [X.sub.[alpha], [beta]] / G [right arrow] G such that M [congruent to] [(M / G).sup.[??]].

(A permutation group G action on [OMEGA] is called fixed-free if [G.sub.x] = [1.sub.G] for [for all] x [member of] [OMEGA].)

[section] 1.11 Semi-Arcs of a Graph

An edge e = uv 2 E([GAMMA]) can be divided into two semi-arcs [e.sub.u], [e.sub.v].

[X.sub.1/2] ([GAMMA])-the set of semi-arcs.

Incidence of Semi-Arcs:

Call u the root vertex in the semi-arc [e.sub.u]. Two semi-arc [e.sub.u], [f.sub.v] are said v-incident or e-incident if u = v or e = f.

[section] 1.12 A Semi-Arc Automorphism

An 1 - 1 mapping [xi] on [X.sub.1/2] ([GAMMA]) such that [for all][e.sub.u], [f.sub.v] [member of] [X.sub.1/2] ([GAMMA]), [xi]([e.sub.u]) and [xi]([f.sub.v]) are v-incident or e-incident if [e.sub.u] and [f.sub.v] are v-incident or e-incident, is called a semi-arc automorphism of the graph [GAMMA].

[Aut.sub.1/2][GAMMA]--the semi-arc automorphism group of [GAMMA]

For [for all]g [member of] Aut[GAMMA], there is also an induced action g|[sup.1/2] on [X.sub.1/2] ([GAMMA]), g : [X.sub.1/2] ([GAMMA]) [right arrow] [X.sub.1/2] ([GAMMA]), as follows:

[for all][e.sub.u] [member of] [X.sub.1/2] ([GAMMA]), g([e.sub.u]) = (g(e)g(u).

All induced action of the elements in Aut[GAMMA] on [X.sub.1/2] ([GAMMA]) is denoted by Aut[GAMMA]|[sup.1/2]. Notice that

Aut[GAMMA] [congruent to] Aut[GAMMA]|[sup.1/2].

Theorem 1.4. For a graph [GAMMA] without loops,

[Aut.sub.1/2][GAMMA] = Aut[GAMMA]|[sup.1/2].

Theorem 1.5. For two maps [M.sub.1] = ([X.sub.[alpha],[beta]], [P.sub.1]) and [M.sub.2] = ([X.sub.[alpha],[beta]], [P.sub.2]) underlying a graph [GAMMA], then

(i) [M.sub.1], [M.sub.2] are equivalent iff [M.sub.1], [M.sub.2] are in one orbits of [Aut.sub.1/2][GAMMA] action on [X.sub.1/2] ([GAMMA]);

(ii) [M.sub.1] [M.sub.2] are isomorphic iff [M.sub.1], [M.sub.2] are in one orbits of [Aut.sub.1/2][GAMMA] x < [alpha] > action on [X.sub.[alpha],[beta]].

Part II Automorphisms of Maps and Klein Surfaces

[section] 2.1 Relation of Maps with Klein Surfaces

Angles incident with a Quadricell:

For a map M = ([X.sub.[alpha],[beta]], P), x [member of] [X.sub.[alpha],[beta]], the permutation pair {(x, Px), ([alpha]x, [P.sup.-1][alpha]x)} is called an angle incident with x.

Theorem 2.1. Any automorphism of a map M = ([X.sub.[alpha][beta]], P) is conformal.

Theorem 2.2. If M is a locally orientable map of genus q, then AutM is isomorphic to a group of comformal transformations of a compact Klein surface of genus q.

(For Riemann surfaces, the same result gotten by Jones and Singerman in 1978.)

[section] 2.2 The Euler Characteristic of Lifting Map

Theorem 2.3. The Euler characteristic X([M.sup.[??]]) of the lifting map [M.sup.[??]] of the voltage map (M, G) is

X ([M.sup.[??]) = |G|(X(M) + [summation over (m [member of [??] (F(M)))] (- 1 + 1 / m)),

where [??](F(M)) denotes the order o(F) set of the faces in M.

[section] 2.3 A Group Being That of a Map

Theorem 2.4 If a group G, G [??] AutM, is fixed-free on V (M), then

|G|(X(M/G) + [summation over (m [member of [??] (F(M/G)))] (-1 + 1 / m)) = X(M).

Corollary 2.1. If M is an orientable map of genus p, G [??] AutM is fixed-free on V (M) and the quotient map M/G with genus [gamma], then

|G| = 2p - 2 / 2[gamma] - 2 + [summation over (m [member of [??] (F(M/G)))] (1 - 1/m)).

Particularly, if M/G is planar, then

|G| = 2p - 2 / -2 + [summation over (m [member of [??] (F(M/G)))] (1 - 1/m)).

Corollary 2.2. If M is a non-orientable map of genus q, G [??] AutM is fixed-free on V (M) and the quotient map M/G with genus [delta], then

|G| = q - 2 / [delta] - 2 + [summation over (m [member of [??] (F(M/G)))] (1 - 1/m)).

Particularly, if M/G is projective planar, then

|G| = q - 2 / -1 + [summation over (m [member of [??] (F(M/G)))] (1 - 1/m)).

Theorem 2.5. If a group G,G [??] AutM, then

X(M) + [summation over (g[member of]G, g[not equal to][1.sub.G]])] (|[[PHI].sub.v](g)| + |[[PHI].sub.f](g)|) = |G|X(M/G),

where, [[PHI].sub.v](g) = {v|v [member of] V (M), [v.sup.g] = v} and [[PHI].sub.f](g) = {f|f [member of] F(M), [f.sup.g] = f}, and if G is fixed-free on V (M), then

X(M) + [summation over (g[member of]G, g[not equal to][1.sub.G]])] |[[PHI].sub.f] (g)| = |G|X(M/G).

Corollary 2.3. If a finite group G, G [??] AutM is fixed-free on V (M) and transitive on F(M), for example, M is regular and G = AutM, then M/G is an one face map and

X(M) = |G|(X(M/G) - 1) + [empty set](M)

Corollary 2.4. For an one face map M, if G, G [??] AutM is fixed-free on V (M), then

X(M) - 1 = |G|(X(M/G) - 1),

and |G|, especially, |AutM| is an integer factor of X(M) - 1.

Remark 2.1. For an one face planar map, i.e., the plane tree, the only fixed-free automorphism group on its vertices is the trivial group by the Corollary 2:4.

[section] 2.4 The Non-Euclid Area of a Map

For a given voltage map (M;G), its non-Euclid area [mu] (M,G) is

[mu](M,G) = 2[pi](-X(M) + [summation over (m [member of [??] (F(M)))] (-1 + 1 / m)).

Particularly, since any map M can be viewed as a voltage map (M, [1.sub.G]), we get the non-Euclid area of a map M

[mu](M) = [mu](M, [1.sub.G]) = -2[pi]X(M).

Theorem 2.6. (Riemann-Hurwitz formula) If G [??] AutM is fixed-free on V (M), then

|G| = [mu](M) / [mu](M/G, [??]).

Theorem 2.7. The non-Euclid area [mu]([DELTA]) of a triangle [DELTA] on a surface S with internal angles [eta], [theta], [sigma] is

[mu]([DELTA]) = [eta] + [theta] + [sigma] - [pi].

[section] 2.5 A Combinatorial Refinement of Huriwtz Theorem

Graphical property P:

Define a graphical property P to be a kind of subgraphs of a graph [GAMMA], such as, regular subgraphs, circuits, trees, stars, wheels, ....

Call a subset A of [X.sub.[alpha],[beta]] of M = ([X.sub.[alpha], [beta]], P) has the graphical property P if the underlying graph of A has property P.

[??](P,M)--the set of all the A subset with property P in the map M.

Theorem 2.8. Let M = ([X.sub.[alpha],[beta]], P) be a map. Then for [for all]G [??] AutM,

[|[v.sup.G][parallel]v [member of] V (M)] | |G|

|G| | |A[parallel][??](P,M)|,

where [a, b, ...] denotes least common multiple of a, b, ....

Corollary 2.5. Let [??] [r.sub.2] be the set of tours with each edge appearing 2 times. Then for [for all]G [??] AutM,

|G| | (l|[??] [r.sub.2]|, l = |T| = |T| / 2 [greater than or equal to] 1, T [member of] [??] [r.sub.2],).

Let [??] [r.sub.1] be the set of tours without repeat edges. Then

|G| | (2l|[??] [r.sub.1]|, l = |T| = |T|/2 [greater than or equal to] 1, T [member of] T [r.sub.1],).

Particularly, denote by [empty set](i, j) the number of faces in M with facial length i and singular edges j, then

|G| |((2i - j)[empty set](i, j), i, j [greater than] 1),

where, (a, b, ...) denotes the greatest common divisor of a, b, ....

Corollary 2.6. Let [??] be the set of trees in the map M. Then for [for all]G [??] AutM,

|G| | (2l[t.sub.l], l [greater than or equal to] 1),

where [t.sub.l] denotes the number of trees with l edges.

Corollary 2.7. Let [v.sub.i] be the number of vertices with valence i. Then for [for all] G [??] AutM,

|G| | (2i[v.sub.i], i [greater than or equal to] 1).

Theorem 2.9. Let M be an orientable map of genus g(M) [greater than or equal to] 2. Then for [for all]G [??] [Aut.sup.+] M,

|G| [less than or equal to] 84(g(M) - 1)

and for [for all]G [??] AutM,

|G| [less than or equal to] 168(g(M) - 1).

Corollary 2.8. For any Riemann surface S of genus g [less than or equal to] 2,

4g(S) + 2 [less than or equal to] |[Aut.sup.+]S| [less than or equal to] 84(g(S) - 1)

8g(S) + 4 [less than or equal to] |AutS| [less than or equal to] 168(g(S) - 1).

Theorem 2.10. Let M be a non-orientable map of genus g'(M) [greater than or equal to] 3. Then for [for all]G [??] [Aut.sup.+]M,

|G| [less than or equal to] 42(g'(M) - 2)

and for [for all]G [??] AutM,

|G| [less than or equal to] 84(g'(M) - 2),

with the equality hold iff M is a regular map with vertex valence 3 and face valence 7 or vice via.

Corollary 2.9. For any Klein surface K underlying a non-orientable surface of genus q [greater than or equal to] 3, |[Aut.sup.+]K| [less than or equal to] 42(q - 2) and |AutK| [less than or equal to] 84(q - 2).

[section] 2.6 The Cyclic Group of a Klein Surface

Theorem 2.11. Let M = ([X.sub.[alpha],[beta]], P) be a map and N = [p.sup.[r.sub.1].sub.1] ... [p.sup.[r.sub.k].sub.k], [p.sub.1] < [p.sub.2] < ... < [p.sub.k], be the arithmetic decomposition of the integer N. Then for any voltage assignment [??] : [X.sub.[alpha],[beta]] [right arrow] [Z.sub.N],

(i) if M is orientable, the minimum genus [g.sub.min] of the lifting map [M.sup.[??]] which admits an automorphism of order N, fixed-free on V ([M.sup.[??]]), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(ii) if M is non-orientable, the minimum genus [g'.sub.min] of the lifting map [M.sup.[??]] which admits an automorphism of order N, fixed-free on V ([M.sup.[??]]), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 2.12. The maximum order [N.sub.max] of an automorphism g of an orientable map M of genus [greater than or equal to] 2 is

[N.sub.max] [less than or equal to] 2g(M) + 1

and the maximum order [N'.sub.max] of anautomorphism g of a non-orientable map of genus, [greater than or equal to] 3 is

[N.sub.'max] [less than or equal to] g(M) + 1,

where g(M) is the genus of the map M.

Corollary 2.10. The maximum order of an automorphism of a Riemann surface of genus [greater than or equal to] 2 is 2g(M) + 1, and the maximum order of an automorphism of a non-orientable Klein surface of genus [greater than or equal to] 3 without boundary is g(M) + 1.

[section] 2.7 The Subgroup of a Graph Being That of Maps

Theorem 2.13. Let [GAMMA] be a connected graph. If G [less than or equal to] Aut[GAMMA], then G is an automorphism group of a map underlying the graph [GAMMA] iff for [for all]v [member of] V([GAMMA]), the stabler [G.sub.v] [less than or equal to] < v > x < [alpha] >.

Theorem 2.14. Let [GAMMA] be a connected graph. If G [less than or equal to] Aut[GAMMA], then G is an orientation-preserving automorphism group of a map underlying the graph [GAMMA] iff for [for all]v [member of] V([GAMMA]), the stabler [G.sub.v] [less than or equal to] < v > is a cyclic group.

Theorem 2.15. Let M be a map underlying the graph G and [o.sub.max](M, g), [o.sub.max](G, g) be the maximum order of orientation-preserving automorphism in AutM and in [Aut.sub.1/2] G. Then

[o.sub.max](M, g) [less than or equal to] [o.sub.max](G, g),

and the equality hold for at least one map underlying the graph G.

Corollary 2.11. For any positive integer n, there exists a vertex transitive map M underlying a circultant such that [Z.sub.n] is an orientation-preserving automorphism group of the map M.

Corollary 2.12. The maximum order of an orientation--preserving automorphism of a complete map [K.sub.n], n [greater than or equal to] 3, is at most n.

Part III The representation of Automorphisms of a Map

[section] 3.1 Complete Maps

A map underlying a complete graph [K.sub.n] is called a complete map. Let [K.sub.n] be a complete graph of order n. Label its vertices by integers 1, 2; ..., n. Then its edge set is {ij|1 [less than or equal to] i, j [less than or equal to] n, i [not equal] j ij = ji}, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Theorem 3.1. All orientation-preserving automorphisms of non-orientable complete maps of order [greater than or equal to] 4 are extended actions of elements in

[[epsilon].sub.[s n/s]], [[epsilon].sub.[1 s n-1/s]],

and all orientation-reversing automorphisms of non-orientable complete maps of order, [greater than or equal to] 4 are extended actions of elements in

[alpha][[epsilon].sub.[(2s) n/2s]], [alpha][[epsilon].sub.[(2s) 4/2s]], [alpha][[epsilon].sub.[1, 1,2]],

where, [[epsilon].sub.[theta]] denotes the conjugatcy class containing element [theta] in the symmetry group [S.sub.n].

Theorem 3.2. All orientation-preserving automorphisms of orientable complete maps of Order [greater than or equal to] 4 are extended actions of elements in

[[epsilon].sub.[s n/s]], [[epsilon].sub.[1,s n-1/s]]

and all orientation-reversing automorphisms of orientable complete maps of order [greater than or equal to] 4 are extended actions of elements in

[alpha][[epsilon].sub.[(2s) n/2s]], [alpha][[epsilon].sub.[(2s) 4/2s]], [alpha][[epsilon].sub.[1, 1,2]],

where, [[epsilon].sub.[theta]] denotes the conjugatcy class containing [theta] in [S.sub.n].

[section] 3.2 Semi-Regular Maps

A graph is semi-regular if it is simple and its automorphism group action on its ordered pair of adjacent vertices is fixed-free and a map is semi-regular if it underlying a semi-regular graph.

Theorem 3.3. Let [GAMMA] be a semi-regular graph. Then all the automorphisms of orientable maps underlying the graph [GAMMA] are

[g|.sup.[X.sub.[alpha],[beta]]] and [alpha]h[|sup.[X.sup.[alpha],[beta]]] , g, h [member of] Aut[GAMMA] with h [equivalent to] 0(mod2).

and all the automorphisms of non-orientable maps underlying the graph [GAMMA] are also

[g|.sup.[X.sub.[alpha],[beta]]] and [alpha]h[|sup.[X.sup.[alpha],[beta]]] , g, h [member of] Aut[GAMMA] with h [equivalent to] 0(mod2).

[section] 3.3 One Face Maps

Theorem 3.4. Let [B.sub.n] be a bouquet with n edges 1, 2,..., n. Then the automorphisms (g; [h.sub.1], [h.sub.2],..., [h.sub.n]) of orientable maps underlying a [B.sub.n], n [greater than or equal to] 1, are respective

(O1) g [member of] [[epsilon].sub.[k n/k]], [h.sub.i] = 1, i = 1, 2,..., n;

(O2) g [member of] [[epsilon].sub.[k n/k]] and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [i.sub.j] [member of] {1, 2,..., n}, n/k [equivalent to] 0(mod2), then [h.sub.[i.sub.1]] = (1, [alpha][beta]), i = 1, 2,..., n/k and [h.sub.[i.sub.j]] = 1 for j [greater than or equal to] 2;

(O3) g [member of] [[epsilon].sub.[[k.sup.2s],(2k) n-2ks/2k]] and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [i.sub.j], [e.sub.[j.sub.t]] [member of] {1, 2,..., n}, then [h.sub.[i.sub.1]] = (1, [alpha][beta]), i = 1, 2,..., s, [h.sub.[i.sub.l]] = 1 for l [greater than or equal to] 2 and [h.sub.[j.sub.t]] = 1 for t = 1, 2,..., 2k and the automorphisms (g; [h.sub.1], [h.sub.2],..., [h.sub.n]) of non-orientable maps underlying a [B.sub.n], n [greater than or equal to] 1, are respective

(N1) g [member of] [[epsilon].sub.[k n/k]], [h.sub.i] = 1, i = 1, 2,..., n;

(N2) g [member of] [[epsilon].sub.[k n/k]], and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [i.sub.j] [member of] {1, 2,..., n}, n/k [equivalent to] 0(mod2), then [h.sub.[i.sub.1]] = (1, [alpha][beta]), (1, [beta]) with at least one [h.sub.[i.sub.01]](1, [beta]), i = 1, 2,..., n/k and [h.sub.[i.sub.j]] = 1 for j [greater than or equal to] 2;

(N3) g [member of] [[epsilon].sub.[[k.sup.2s], (2k) [n-2ks/2k]] = and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [i.sub.j], [e.sub.[j.sub.t]] [member of] {1, 2,..., n}, then [h.sub.[i.sub.1]] = (1 [alpha][beta]), (1, [beta])) with at least one [h.sub.[i.sub.01]] = (1, [beta]), i = 1, 2,..., s, [h.sub.il] = 1 for l [greater than or equal to] 2 and [h.sub.[j.sub.t]] = 1 for t = 1, 2,..., 2k.

Part IV The Enumeration of Unrooted Maps

[section] 4.1 A Scheme for Enumeration

Theorem 4.1. For a given graph [GAMMA], let [epsilon] [subset] [[epsilon].sup.L]([GAMMA]), then the numbers n([epsilon],[GAMMA]) and [eta]([epsilon], [GAMMA]) of non-isomorphic unrooted maps and non-equivalent embeddings in [epsilon] are respective

n([epsilon], [GAMMA]) = 1/2|[Aut.sub.1/2] [GAMMA]| [summation over (g [member of] [Aut.sub.1/2] [GAMMA])] |[[PHI].sub.1] (g)|,

where, [[PHI].sub.1] (g) = {P|P [member of] [epsilon] [epsilon] and [P.sup.g] = P or [P.sup.g[alpha]] = P} and

[eta]([epsilon], [GAMMA]) = 1|[Aut.sub.1/2] [GAMMA]| [summation over (g [member of] [Aut.sub.1/2] [GAMMA])] |[[PHI].sub.2] (g)|,

where, [[PHI].sub.2](g) = {P|P [member of] [epsilon] and [P.sup.g] = P}.

Corollary 4.1. The numbers [n.sup.O] ([GAMMA]), [n.sup.N]([GAMMA]) and [n.sup.L]([GAMMA]) of non-isomorphic unrooted orientable maps ,non-orientable maps and locally orientable maps underlying a graph [GAMMA] are respective

[n.sup.O]([GAMMA]) = 1/2|[Aut.sub.1/2] [GAMMA]| [summation over (g [member of] [Aut.sub.1/2] [GAMMA])] |[[PHI].sup.O.sub.1] (g)|;

[n.sup.N] (T) = 1/2|[Aut.sub.1/2] [GAMMA]| [summation over (g [member of] [Aut.sub.1/2] [GAMMA])] |[[PHI].sup.N.sub.1] (g)|;

[n.sub.L] ([GAMMA]) = 1/2|[Aut.sub.1/2] [GAMMA]| [summation over (g [member of] [Aut.sub.1/2] [GAMMA])] |[[PHI].sup.L.sub.1] (g)|,

where, [[PHI].sup.O.sub.1](g) = {P|P [member of] [[epsilon].sup.O] ([GAMMA]) and [P.sup.g] = P or [P.sup.g[alpha]] = P}, [[PHI].sup.N.sub.1](g) = {P|P [member of] [[epsilon.N]([GAMMA]) and [P.sup.g] = P or [P.sup.g[alpha]] = P}, [[PHI].sup.L.sub.1](g) = {P|P [member of] [[epsilon].sup.L]([GAMMA]) and [P.sup.g] = P or [P.sup.g[alpha]] = P}.

[section] 4.2 The Number of Complete Maps

Theorem 4.2. The number [n.sup.L]([K.sub.n]) of complete maps of order n [greater than or equal to] 5 on surfaces is

[n.sup.L]([K.sub.n]) = 1/2 ([summation over (k|n)] + [summation over (k|n,k[equivalent to]0(mod2))]) [2.sup.[alpha](n,k)] [(n - 2)!.sup.n/k]/[k.sup.n/k](n/k)! + [summation over (k|(n-1),k[not equal to]1)] [empty set](k)[2.sup.[beta](n,k)] [(n - 2)!.sup.n-1/k/n-1,

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

and [n.sup.L]([K.sub.4]) = 11.

Theorem 4.3. The number [n.sup.O](([K.sub.n]) of complete maps of order n [greater than or equal to] 5 on orientable surfaces is

[n.sup.O]([K.sub.n]) = 1/2 ([summation over (k|n)] + [summation over (k|n,k[equivalent to]0(mod2))]) [(n - 2)!.sup.n/k]/[k.sup.n/k](n/k)! + [summation over (k|(n-1),k[not equal to]1)] [empty set](k) [(n -2)!.sup.n-1/k/n-1.

and n([K.sub.4]) = 3: For [K.sub.4] on the surfaces, see the Fig.2

[FIGURE 2 OMITTED]

[section] 4.3 The Number of Semi-Regular Maps

Theorem 4.4. Let [GAMMA] be a semi-regular graph. Then the numbers of unrooted maps on orientable and non-orientable surfaces underlying the graph [GAMMA] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]!

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]!,

where [lambda]([xi]) = 1 if o([xi]) [equivalent to] 0(mod2) and 1/2, otherwise.

Corollary 4.2. Let [GAMMA] = Cay([Z.sub.p] : S) be connected graph of prime order p with (p-1, |S|) = 2. Then

[n.sup.O]([GAMMA], M) = [(|S| - 1)!.sup.p] + 2p[(|S| - 1)!.sup.p+1/2]/4p + (p - 1)(|S| - 1)!/4p

and

[n.sup.N]([GAMMA], M) = ([2.sup.p|S|/2 -p] - 1)[(|S| - 1)!.sup.p]/2p + 2([2.sup.p|S|-2p-2)/4] -1)p [(|S| - 1)!.sup.p+1/2]/2p + ([2.sup.|S|-2/2] - 1)(p -1) (|S| - 1)!/4p.

[section] 4.4 The Number of One Vertex Maps

Theorem 4.5. The number [n.sup.O]([B.sub.n]) of non-isomorphic maps on orientable surfaces underlying a graph [B.sub.n] is

[n.sup.O]([B.sub.n]) = [summation over (k|2n,k[not equal to]2n)] [k.sup.2n/k -1] (2n/k - 1)! 1/(2n/k)! [[partial derivative].sup.2n/k] (Z([S.sub.n][[S.sub.2]]))/[partial derivative][s.sup.2n/k.sub.k] | [s.sub.k] = 0

+ [empty set](2n) [partial derivative](Z([S.sub.n][[S.sub.2]]))/[partial derivative][s.sub.2n]| [s.sub.2n]=0

Theorem 4.6. he number [n.sup.N]([B.sub.n]) of non-isomorphic maps on the non-orientable surfaces with an underlying graph [B.sub.n] is

[n.sup.N]([B.sub.n]) = (2n - 1)!/n! + [summation over (k|2n,3[less than or equal to]k<2n)] [(2k).sup.2n/k -1] (2n/k - 1)! [[partial derivative].sup.2n/k] (Z([S.sub.n][[S.sub.2]]))/[partial derivative][s.sup.2n/k.sub.k] | [s.sub.k] = 0 + 1/[2.sup.n]n! ([summation over (s[greater than or equal to]1)] n!/(n - 2s)!s! + [4.sup.n] (n -1)! [[partial derivative].sup.n] (Z([S.sub.n][[S.sub.2]]))/[partial derivative][s.sup.n.sub.2] | [s.sub.2]=0 - [n/2])).

For [B.sub.2] on the surfaces, see the Fig.3.

[FIGURE 3 OMITTED]

Part V Map Geometry

[section] 5.1 What are the Contribution of Maps to Mathematics Klein Erlanger Program:

Any geometry is finding invariant properties under the transformation group of this geometry (This is essentially the group action idea.)

The following problems are applications of the Klein Erlanger Program in maps:

(i) determine isomorphism maps or rooted maps;

(ii) determine equivalent embeddings of a graph;

(iii) determine an embedding whether exists;

(iv) enumerate maps or rooted maps on a surface;

(v) enumerate embeddings of a graph on a surface;

(vi) ..., etc.

What are their importance to classical mathematics?

What are their contributions to science? [section] 5.2 Smarandache Geometries

Classical geometries:

The axiom system of Euclid geometry is the following:

(A1) there is a straight line between any two points.

(A2) a finite straight line can produce a infinite straight line continuously.

(A3) any point and a distance can describe a circle.

(A4) all right angles are equal to one another.

(A5) if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The axiom (A5) can be also replaced by:

(A5') given a line and a point exterior this line, there is one line parallel to this line.

The Lobachevshy-Bolyai-Gauss geometry, also called hyperbolic geometry, is a geometry with axioms (A1)-(A4) and the following axiom (L5):

(L5) there are infinitely many line parallels to a given line passing through an exterior point.

The Riemann geometry is a geometry with axioms (A1)-(A4) and the following axiom (R5):

there is no parallel to a given line passing through an exterior point.

Smarandache introduced the paradoxist geometry, non-geometry, counter-projective geometry and anti-geometry by contradicts the axioms (A1)-(A5) in Euclid geometry, generalize the classical geometries. For example, the axioms of a Paradoxist geometry are (A1)-(A4) and with one of the following as the axiom (P5):

(i) there are at least a straight line and a point exterior to it in this space for which any line that passes through the point intersect the initial line.

(ii) there are at least a straight line and a point exterior to it in this space for which only one line passes through the point and does not intersect the initial line.

(iii) there are at least a straight line and a point exterior to it in this space for which only a finite number of lines [l.sub.1], [l.sub.2],..., [l.sub.k], k [greater than or equal to] 2 pass through the point and do not intersect the initial line.

(iv) there are at least a straight line and a point exterior to it in this space for which an infinite number of lines pass through the point (but not all of them) and do not intersect the initial line.

(v) there are at least a straight line and a point exterior to it in this space for which any line that passes through the point and does not intersect the initial line.

F. Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

The Smarandache geometries are defined as follows.

Definition 5.1. An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969).

A nice model for the Smarandache geometries, called s-manifolds, is found by Isier, which is defined by Mao using maps as follows:

An s-manifold is any collection C(T, n) of these equilateral triangular disks [T.sub.i], 1 [less than or equal to] i [less than or equal to] n satisfying the following conditions:

(i) Each edge e is the identification of at most two edges [e.sub.i], [e.sub.j] in two distinct triangular disks [T.sub.i], [T.sub.j], 1 [less than or equal to] i, j [less than or equal to] n and i [not equal to] j;

(ii) Each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks.

H.Iseri, Smarandache manifolds, American Research Press, Rehoboth, NM,2002.

L.F.Mao, Automorphism groups of maps, surfaces and Smarandache geometries, American Research Press, Rehoboth, NM,2005.

[section] 5.3 A Classification of Smarandache Manifolds

Classical Type:

(1) [[DELTA].sub.1] = {5--regular triangular maps} (elliptic);

(2) [[DELTA].sub.2] = {6--regular triangular maps} (euclidean);

(3) [[DELTA].sub.3] = {7--regular triangular maps} (hyperbolic).

Smarandache Type:

(4) [[DELTA].sub.4] = {triangular maps with vertex valency 5 and 6} (euclid-elliptic);

(5) [[DELTA].sub.5] = {triangular maps with vertex valency 5 and 7} (elliptic-hyperbolic);

(6) [[DELTA].sub.6] = {triangular maps with vertex valency 6 and 7} (euclid-hyperbolic);

(7) [[DELTA].sub.7] = {triangular maps with vertex valency 5, 6 and 7} (mixed).

Theorem 5.1. |[[DELTA].sub.1]| = 2, |[[DELTA].sub.5]| [greater than or equal to] 2 and |[[DELTA].sub.i]|, i = 2, 3, 4, 6, 7 are infinite.

Iseri proposed a question: Do the other closed 2-manifolds correspond to s-manifolds with only hyperbolic vertices?. Since |[[DELTA].sub.3]| is infinite, the answer is affirmative for this question.

[section] 5.4 Map Geometry

Definition 5.2. For a combinatorial map M with each vertex valency [greater than or equal to] 3, associates a real number [mu](u); 0 < [mu](u) [pi], to each vertex u; u [member of] V (M). Call (M, [mu]) the fundamental map space, [mu](u) the angle factor of the vertex u and to be orientable or non-orientable if the map M is orientable or not.

Definition 5.3. A point u in a map space (M, [mu]) is called elliptic, euclidean or hyperbolic if [rho](u)[mu](u) < 2[pi], [rho](u)[mu](u) = 2[pi] or [rho](u)[mu](u) > 2[pi].

Definition 5.4. Let (M, [mu]) be a map space. An m-line in (M, [mu]) is a curve with a constant curvature. Points in (M, 1) are called m-points.

We have the following result for map geometries.

Theorem 5.2. For any planar map M with order [greater than or equal to] 3 and vertex valency [greater than or equal to] 3, there is an angle factor [mu] such that (M, [mu]) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5), (L5) and (R5).

[FIGURE 4 OMITTED]

Theorem 5.3. For any map M on an orientable surface with order [greater than or equal to] 3 and vertex valency [greater than or equal to] 3, there is an angle factor [mu] such that (M, [mu]) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

Theorem 5.4. Let P be a k-polygon in a map space with each line segment passes through at most one elliptic or hyperbolic point. If H is the set of elliptic points and hyperbolic points on the line segment of P, then the sum of the internal angles in P is

(k + |H| - 2)[pi] - 1/2 [summation over (u[member of]H)] [rho](u)[mu](u).

Corollary 5.1. Let [DELTA] be a triangle in a map space. Then

(i) if [DELTA] is euclidean, then then the sum of its internal angles is equal to [pi];

(ii) if [DELTA] is elliptic, then the sum of its internal angles is less than [pi];

(iii) if [DELTA] is hyperbolic, then the sum of its internal angles is more than [pi].

Theorem 5.5. The number [n.sup.O]([GAMMA], g) of non -equivalent orientable map geometries underlying a simple graph [GAMMA] by denial the axiom (A5) by (A5), (L5) or (R5) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [rho](v) is the valency of the vertex v in the graph G.

Part VI Open Problems for Combinatorial Maps

[section] 6.1 The Uniformization Theorem for Simple Connected Riemann Surfaces

The uniformization theorem for simple connected Riemann surfaces is one of those beautiful results in the Riemann surface theory, which is stated as follows:

If S is a simple connected Riemann surface, then S is conformally equivalent to one and only one of the following three:

(a) C [union] [infinity];

(b) C;

(c) [DELTA] = {z [member of] C[parallel]|z| < 1}.

How can we define the conformal equivalence for maps enabling us to get the uniformization theorem of maps?

What is the correspondence class maps with the three type (a)-(c) Riemann surfaces?

[section] 6.2 Combinatorial Construction of an Algebraic Curve of Genus

A complex plane algebraic curve [C.sub.l] is a homogeneous equation f(x, y, z) = 0 in [P.sub.2]C = ([C.sup.2] \ (0, 0, 0))/ ~, where f(x, y, z) is a polynomial in x, y and z with coefficients in C. The degree of f(x, y, z) is said the degree of the curve [C.sub.l]. For a Riemann surface S, a well-known result is ([2]) there is a holomorphic mapping [phi] : S [right arrow] [P.sub.2]C such that [phi](S) is a complex plane algebraic curve and

g(S) = (d([phi](S)) - 1)(d([phi](S)) - 2)/2 .

By map theory, we know a combinatorial map also is on a surface with genus. Then whether we can get an algebraic curve by all edges in a map or by make operations on the vertices or edges of the map to get plane algebraic curve with given k-multiple points?

how do we find the equation f(x, y, z) = 0?

[section] 6.3 Classification of s-Manifolds by Maps

We present an elementary classification for the closed s-manifolds in the Part V. For the general s-manifolds, their correspondence combinatorial model is the maps on surfaces with boundary, founded by Bryant and Singerman in 1985 (R.P.Bryant and D.Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford(2),36(1985), 17-41.). The later is also related to the modular groups of spaces and need to investigate further itself. The questions are

(i) how can we combinatorially classify the general s-manifolds by maps with boundary?

(ii) how can we find the automorphism group of an s-manifold?

(iii) how can we know the numbers of non-isomorphic s-manifolds, with or without root?

(iv) find rulers for drawing an s-manifold on a surface, such as, the torus, the projective plane or Klein bottle, not the plane.

[section] 6.4 Map Geometries

(i) For a given graph, determine properties of the map geometries underlying this graph.

(ii) For a given locally orientable surface, determine the properties of map geometries on this surface.

(iii) Classify map geometries on a locally orientable surface.

(iv) Enumerate non-equivalent map geometries underlying a graph or on a locally orientable surface.

(v) Establish the surface geometry by map geometries.

(vi) Applying map geometries to classical mathematics or other sciences.

[section] 6.5 Gauss Mapping Among Surfaces

In the classical differential geometry, a Gauss mapping among surfaces is defined as follows:

Let S [subset] [R.sup.3] be a surface with an orientation N. The mapping N : S [right arrow] [R.sup.3] takes its value in the unit sphere

[S.sup.2] = {(x, y, z) [member of] [R.sup.3]|[x.sup.2] + [y.sup.2] + [z.sup.2] = 1}

along the orientation N. The map N : S [right arrow] [S.sup.2], thus defined, is called the Gauss mapping.

we know that for a point P [member of] S such that the Gaussian curvature K(P) [not equal to] 0 and V a connected neighborhood of P with K does not change sign,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where A is the area of a region B [subset] V and N(A) is the area of the image of B by the Gauss mapping N : S [right arrow] S2. The questions are

(i) what is its combinatorial meaning of the Gauss mapping? How to realizes it by maps?

(ii) how we can define various curvatures for maps and rebuilt the results in the classical differential geometry?

[section] 6.6 The Gauss-Bonnet Theorem

Let S be a compact orientable surface. Then

[integral] [[integral].sub.S] Kd[sigma] = 2[[pi].sub.[chi]](S),

where K is Gaussian curvature on S.

This is the famous Gauss-Bonnet theorem for compact surface. The questions are

(i) what is its combinatorial mean of the Gauss curvature?

(ii) how can we define the angle, area, volume, curvature,..., of a map?

(iii) can we rebuilt the Gauss-Bonnet theorem by maps? or can we get a generalization of the classical Gauss-Bonnet theorem by maps?

(1) Reported at the Academy of Mathematics and Systems of Chinese Academy of Sciences.

Linfan Mao

Institute of Systems, Academy of Mathematics and System Sciences,

Chinese Academy of Sciences, Beijing, P.R. China
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