Enumeration of unrooted maps of a given genus

Let N g ( f ) denote the number of rooted maps of genus g having f edges. An exact formula for N g ( f ) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2 , 3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number Θ γ ( e ) of unrooted maps on an orientable surface S γ of a given genus γ and with a given number of edges e. It has a form of a linear combination ∑ i , j c i , j N g j ( f i ) of numbers of rooted maps N g j ( f i ) for some g j ⩽ γ and f i ⩽ e . The coefficients c i , j are functions of γ and e. We consider the quotient S γ / Z ℓ of S γ by a cyclic group of automorphisms Z ℓ as a two-dimensional orbifold O. The task of determining c i , j requires solving the following two subproblems:(a) pute the number Epi o ( Γ , Z ℓ ) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O = S γ / Z ℓ onto Z ℓ ; culate the number of rooted maps on the orbifold O which lifts along the branched covering S γ → S γ / Z ℓ to maps on S γ with the given number e of edges. mber Epi o ( Γ , Z ℓ ) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers N g ( f ) for some g ⩽ γ and f ⩽ e . It follows that Θ γ ( e ) can be calculated whenever the numbers N g ( f ) are known for g ⩽ γ and f ⩽ e . In the end of the paper the above approach is applied to derive the functions Θ γ ( e ) explicitly for γ ⩽ 3 . We note that the function Θ γ ( e ) was known only for γ = 0 (Liskovets, 1981). Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus γ = 1 , 2 , 3 are presented.