Wythoff polytopes and low-dimensional homology of Mathieu groups

Abstract

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as H₅(M₂₃,)=₇ and H₃(M₂₄,)=₁₂. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free M_n-resolution. Both methods apply in principle to arbitrary finite groups.