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WYTHOFF POLYTOPES AND LOWDIMENSIONAL HOMOLOGY OF MATHIEU GROUPS
, 2009
"... We describe two methods for computing the lowdimensional integral homology of the Mathieu simple groups and use them to make computations such as H5(M23, Z) = Z7 and H3(M24, Z) = Z12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to pr ..."
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We describe two methods for computing the lowdimensional integral homology of the Mathieu simple groups and use them to make computations such as H5(M23, Z) = Z7 and H3(M24, Z) = Z12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free ZMnresolution. Both methods apply in principle to arbitrary finite groups.
Hypercube Embeddings of Wythoffians
 ARS MATHEMATICA CONTEMPORANEA 1 (2008) 99–111
, 2008
"... The Wythoff construction takes a ddimensional polytope P, a subset S of {0,..., d} and returns another ddimensional polytope P (S). If P is a regular polytope, then P (S) is vertextransitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We w ..."
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The Wythoff construction takes a ddimensional polytope P, a subset S of {0,..., d} and returns another ddimensional polytope P (S). If P is a regular polytope, then P (S) is vertextransitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want to determine, which of those Wythoffians P (S) with regular P have their skeleton or dual skeleton isometrically embeddable into the hypercubes Hm and halfcubes 1 2 Hm. We find six infinite series, which, we conjecture, cover all cases for dimension d> 5 and some sporadic cases in dimension 3 and 4 (see Tables 1 and 2). Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable P (S) are addressed throughout the text.
WYTHOFF POLYTOPES AND LOWDIMENSIONAL HOMOLOGY OF MATHIEU GROUPS
"... Abstract. We describe two methods for computing the lowdimensional integral homology of the Mathieu simple groups and use them to make computations such as H5(M23,Z) = Z7 and H3(M24,Z) = Z12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation technique ..."
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Abstract. We describe two methods for computing the lowdimensional integral homology of the Mathieu simple groups and use them to make computations such as H5(M23,Z) = Z7 and H3(M24,Z) = Z12. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free ZMnresolution. Both methods apply in principle to arbitrary finite groups. 1.