Enumeration of unrooted maps of a given genus

Let Ng(f) denote the number of rooted maps of genus g having f edges. An 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 a given genus γ and with a given number of edges e. It has a form of a linear combination ∑i,jci,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. We consider the quotient Sγ/Zℓ of Sγ by a cyclic group of automorphisms Zℓ as a two-dimensional orbifold O. The task of determining ci,j requires solving the following two subproblems:(a)to compute the number Epio(Γ,Zℓ) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O=Sγ/Zℓ onto Zℓ;(b)to calculate the number of rooted maps on the orbifold O which lifts along the branched covering Sγ→Sγ/Zℓ to maps on Sγ with the given number e of edges.The number Epio(Γ,Zℓ) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining 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 to derive the functions Θγ(e) explicitly for γ⩽3. We note that the function Θγ(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.