Enumeration of unrooted maps with given genus

Let Ng(f) denote the number of rooted maps of genus g having f edges. Exact formula for Ng(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 Θγ(e) of unrooted maps on an orientable surface Sγ of given genus γ and given number of edges e. It has a form of a linear combination ∑ i,j ci,jNgj (fi) of numbers of rooted maps Ngj (fi) for some gj ≤ γ and fi ≤ e. The coefficients ci,j are functions of γ and e. Let us consider the quotient Sγ/Z` of Sγ by a cyclic group of automorphisms Z` as a two-dimensional orbifold O. The task to determine ci,j requires to solve the following two subproblems: (a) to compute the number Epio(Γ, Z`) of order preserving epimorphisms from the fundamental group Γ of the orbifold O = Sγ/Z` onto Z`, (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering Sγ → Sγ/Z` to maps on Sγ with the given number e of edges. The number Epio(Γ, Z`) is expressed in terms of classical number theoretical functions. The other problem is reduced onto the standard enumeration problem to determine the numbers Ng(f) for some g ≤ γ and f ≤ e. It follows that Θγ(e) can be calculated whenever the numbers Ng(f) are known for g ≤ γ and f ≤ e. In the end of the paper the above approach is applied do derive the functions Θγ(e) explicitly for γ ≤ 3. Let us remark that the function Θγ(e) was known only for γ = 0 (Liskovets 1981). Tables containing the numbers of isomorphism classes of maps up to 30 edges for genus γ = 1, 2, 3 are produced.