Regular hypermaps.

*(English)*Zbl 0665.57002This 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.

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 |

##### Keywords:

decomposition of an orientable surface with the edges and vertices homeomorphic to closed discs; topological hypermaps; algebraic hypermaps; hypervertices; hyperedges; hyperfaces; genus; universal topological hypermap; universal algebraic hypermap
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\textit{D. Corn} and \textit{D. Singerman}, Eur. J. Comb. 9, No. 4, 337--351 (1988; Zbl 0665.57002)

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