Europ. J. Combinatorics (2000) 21, 333–345 Article No. 10.1006/eujc.1999.0332 Available online at http://www.idealibrary.com on Regular Cyclic Coverings of the Platonic Maps GARETH A. JONES†‡ AND DAVID B. SUROWSKI† We use homological methods to describe the regular maps and hypermaps which are cyclic cover- ings of the Platonic maps, branched over the face centers, vertices or midpoints of edges. c 2000 Academic Press 1. INTRODUCTION The Mobius–Kantor¨ map 4 4; 3 [6, Sections 8.8 and 8.9] is a regular orientable map of type 8; 3 and genus 2. Itf isC a 2-sheetedg covering of the cube 4; 3 , branched over the centers off itsg six faces, each of which lifts to an octagonal face. Itsf (orientation-preserving)g automorphism group is isomorphic to GL2.3/, a double covering of the automorphism group PGL2.3/ ∼ S4 of the cube. The aim of this note is to describe all the regular maps and hypermapsD which can be obtained in a similar manner as cyclic branched coverings of the Platonic maps , with the branching at the face centers, vertices, or midpoints of edges. The method usedM is to consider the action of Aut on certain homology modules; in a companion paper [16] we use cohomological techniquesM to give explicit constructions of these coverings in terms of voltage assignments. Conder and Everitt [2] have used different methods to construct non-orientable regular maps as cyclic coverings of smaller maps, branched over their face centers. Let be a Platonic map, that is, a regular map on the sphere S2. In the notation of [6], hasM type n; m , or simply n; m , where the faces are n-gons and the vertices have valencyM m; asf a hypermap,g MhasD type f .mg; 2; n/. Here, 0 .m 2/.n 2/ < 4, so is the dihedron n; 2 , the hosohedronM 2; m , the tetrahedron 3≤; 3 , the− cube−4; 3 , the octahedronM 3; 4 , thef dodecahedrong 5; 3 orf theg icosahedron 3; 5 f. g f g f Weg first determine the regularf g maps which aref d-sheetedg coverings of , with a cyclic group of covering transformations, branchedN over the face centers. Thus Mis a map of type dn; m , or equivalently, a hypermap of type .m; 2; dn/, and the group GN Aut has a Q normalf g subgroup D C with =D , so that G=D G Aut .VD Our mainN result ∼ d ∼ Q ∼ is: D N D M D VD M THEOREM 1. The isomorphism classes of d-sheeted regular cyclic coverings of , N M branched over the face centers, are in one-to-one correspondence with the solutions u Zd of 2 h 2 f 1 u 1 and 1 u u u − 0; . / D C C C···C D ∗ where h hcf .m; 2/ and has f faces. They have type dn; m and genus .d 1/. f 2/=2, and are allD reflexible. TheM group G Aut has a presentationf g − − Q D N x; y; z xm y2 zdn xyz 1;.zn/x znu ; h j D D D D D i with G Aut G=D where D zn . D M D∼ Q D h i †The authors thank Steve Wilson for organizing the workshop SIGMAC 98 (Flagstaff, AZ.), where this collaboration began as a result of the second author’s talk [15]. ‡Author to whom correspondence should be addressed 0195–6698/00/030333 + 13 $35.00/0 c 2000 Academic Press 334 G. A. Jones and D. B. Surowski When u 1 (equivalently, D is in the center of G), we obtain Sherk’s maps d n; m [14], Q one for eachDd dividing f ; if m is odd these are the only possibilities, but if m isf even· weg also obtain non-central cyclic coverings for certain values of d, including d in some cases. By duality, a similar process yields those d-sheeted regular cyclic coversD 1of which are branched over the vertices; these are maps of type n; dm . Finally, if weN allowM branching over the edges of we obtain d-sheeted coverings f whichg are hypermaps (but not maps) M h N 2 e 1 of type .m; 2d; n/, the relevant conditions being u 1; 1 u u u − 0 where h hcf.m; n/ and has e edges; is reflexible exceptD forC theC caseC··· C3; 3 withD u 1. D M N M D f g 6D 2.PRELIMINARIES First we briefly sketch the connections between maps, hypermaps (always assumed to be orientable) and triangle groups; for the details, see [7–9], and for background on hypermaps, see [5]. We define a triangle group to be 1.p; q; r/ x; y; z x p yq zr xyz 1 ; D h j D D D D i where p; q; r N and we ignore any relation g1 1. Any m-valent map corre- sponds to a subgroup2 [ f1gN of the triangle group D N 1 1.m; 2; / x; y; z xm y2 xyz 1 ; VD 1 D h j D D D i with vertices, edges and faces corresponding to the cycles of x; y and z on the cosets of N. The map is regular if and only if N is normal in 1, in which case Aut ∼ 1=N. In particular,N the Platonic map n; m corresponds to the normal closureN DM of zn M D f g in 1, and Aut ∼ 1=M ∼ 1.m; 2; n/. The regular map is a d-sheeted covering of if and onlyM if ND is a subgroupD of index d in M, in whichN case the group of covering transformationsM is M=N. The regular m-valent maps which are d-sheeted cyclic coverings of are therefore in bijective correspondence with the subgroups N of M which are normal in M 1, with M=N Cd . Since M=N is abelian and has exponent d, such subgroups N contain D∼ d the commutator subgroup M0 and the subgroup M generated by the dth powers in M, so they d d correspond to subgroups N N=M0 M of M M=M0 M . The action of 1 by conjugation D D d on the normal subgroup M preserves its characteristic subgroups M0 and M , so there is an induced action of 1 on M; since M is in the kernel of this action, we therefore have an action of the group G Aut 1=M on M, which is a module for G over the ring Zd ; it follows D M D∼ d that a subgroup N of M, containing M0 M , is normal in 1 if and only if N is a G-invariant submodule of M. 2 Let S S c1;:::; c f , where c1;:::; c f are the centers of the f faces of . Then M D n f g M can be identified with the fundamental group π1.S/ of S, a free group of rank f 1 generated − by the homotopy classes gi of loops around the punctures ci , with a single defining relation ab g1 ::: g f 1. It follows that the group M M=M0 can be identified with the first integer D ab D homology group H1.S Z/ π1.S/ of S, a free abelian group of rank f 1 generated by the I D − homology classes gi with g1 g f 0, and then by the universal coefficient theo- T U T UC···CT UD f 1 rem, M is identified with the mod .d/ homology group H1.S Zd / H1.S Z/ Zd ∼ Zd − . Under these identifications, the actions of G induced by conjugationI D in 1 andI by⊗ homeomor-D phisms of S are the same, so our problem is to find the G-submodules of H1.S Zd / of codi- mension 1, or equivalently, the kernels of G-epimorphisms onto one-dimensionalI G-modules. Regular abelian coverings of maps (and more generally Riemann surfaces) with automor- phism group G can be determined by using ordinary and modular representation theory to study the decomposition of the G-module H1.S/ over various rings and fields of coefficients; Regular cyclic coverings 335 see [10, 11, 13] for examples of this technique. In our case, since we are interested only in one-dimensional constituents, we can adopt a rather simpler, more direct approach. Let P be the permutation module over Zd for the action of G on the faces of . As a Zd - M module, this has a basis e1;:::; e f in one-to-one correspondence with the faces, and these are permuted in the same way as G permutes the faces. Now e1 e f generates a G-invariant C···C one-dimensional submodule P1 of P, and H1.S Zd / is isomorphic to the quotient G-module I P=P1. We therefore need to find the G-submodules of codimension 1 in P containing P1, or equivalently, the G-epimorphisms θ P Q where Q is one-dimensional and P1 ker θ. V ! ≤ 3. CENTRAL CYCLIC COVERINGS We first consider the case where D is central in G, so that G acts trivially on Q. The Q resulting maps were described by Sherk [14] from a rather different point of view, but for completenessN we will show how they arise from the above general theory. For notational convenience, we let Z denote Z. 1 LEMMA 2. Let G be a transitive permutation group of degree f , let P be its permutation module over Zd .d N /, with basis e1;:::; e f permuted by G, let P1 be the one- 2 [ f1g dimensional G-submodule of P spanned by ei , and let Q be a one-dimensional G-module P with the trivial action of G.