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Regular hypermaps. (English) Zbl 0665.57002
This paper studies the correspondence between topological hypermaps and algebraic hypermaps. A topological map is a decomposition of a surface into polygonal cells. A topological hypermap is a similar decomposition of an orientable surface with the “edges” and “vertices” (hyperedges and hypervertices) homeomorphic to closed discs. More formally it is a triple $${\mathcal H}=(X,S,A)$$ where X is an orientable surface, S and A have components (hypervertices and hyperedges) homeomorphic to closed discs, $$B=S\cap A$$ (bits) is a finite set for X compact or satisfies some locally finite conditions for noncompact X, and the components (hyperfaces) of $$X\setminus (S\cup A)$$ are homeomorphic to open discs. $${\mathcal H}$$ is of type ($$\ell,m,n)$$ if $$\ell$$, m, and 2n are the least common multiples of the number of bits on the hypervertices, hyperedges and hyperfaces. The genus of $${\mathcal H}$$ is the genus of X.
An algebraic hypermap is a quadruple $${\mathcal A}=(G,B,\sigma,\alpha)$$ where B is a set and $$\sigma$$, $$\alpha$$ are permutations of B such that $$G=gp<\sigma,\alpha >$$ is transitive on B. The type of $${\mathcal A}$$ is ($$\ell,m,n)$$ where $$\ell$$, m and n are the orders of $$\sigma$$, $$\alpha$$, and $$\sigma$$ $$\alpha$$ respectively. The genus, g, of $${\mathcal A}$$ is given by $$z(\sigma)+z(\alpha)+z(\sigma \alpha)=| B| +2-2g$$ where z($$\zeta)$$ is the number of cycles in the permutation $$\zeta$$.
Given a topological hypermap $${\mathcal H}$$, an algebraic hypermap, Alg($${\mathcal H})$$, is constructed by letting B be the set of bits and $$\sigma$$ (resp. $$\alpha)$$ the permutation of B whose cycles are obtained by going around the hypervertices (resp. the hyperedges) in a positive sense. The permutation $$\sigma$$ $$\alpha$$ consists of cycles going around hyperfaces in a negative direction traversing two edges at a time (one from a hyperedge and one from a hypervertex). The construction of Alg($${\mathcal H})$$ preserves type and genus. Isomorphisms of topological and algebraic hypermaps are defined in canonical ways. An algebraic hypermap is regular if Aut $${\mathcal A}$$ is transitive on B.
Universal algebraic and topological hypermaps of type ($$\ell,m,n)$$ are constructed with the property that any hypermap of type ($$\ell,m,n)$$ is a quotient of the appropriate universal object. It is shown that if $$\hat {\mathcal H}$$ is the universal topological hypermap of type ($$\ell,m,n)$$ then Alg($$\hat {\mathcal H})$$ is the universal algebraic hypermap of type ($$\ell,m,n)$$. The universal objects are used to show that every algebraic hypermap can be obtained from a topological hypermap.
Finally regular hypermaps are described for genus $$\leq 2$$. There is a method for constructing regular hypermaps from a regular map of type (2,m,n). Five of the regular hypermaps of genus 2 are not obtained by that construction.
Reviewer: G.Lang

##### MSC:
 57M15 Relations of low-dimensional topology with graph theory 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 54B15 Quotient spaces, decompositions in general topology
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