Applications of the Hopf trace formula to computing homology representations

The primary aim of this paper is to illustrate the use of a well-known technique of algebraic topology, the Hopf trace formula, as a tool in computing homology representations of posets. Inspired by a recent paper of Bjj orner and Lovv asz ((BL]), we apply this tool to derive information about the homology representation of the symmetric group S n on a class of subposets of the partition lattice n : The majority of these subposets are not Cohen-Macaulay. Techniques for such computations have taken on an added signiicance because of recent developments in the theory of subspace arrangements, in particular the equivariant Goresky-MacPherson formula of SWe1]. The latter formula reduces the calculation of the representation on the cohomology of the complement of a subspace arrangement to the problem of computing the homology representation on the intersection lattice of the arrangement. Other methods for computing homology representations of posets arising from the partition lattice were introduced in Su], and were applied to the k-equal partition lattice ((SWa]) and the corresponding subspace arrangement ((SWe1]). Throughout the paper we will assume the reader is familiar with the ordinary representation theory of nite groups. The rst two sections of this paper are written with the non-specialist in mind. (Others may want to proceed directly to the discussion following Corollary 2.8.) Section 1 contains an elementary exposition of the Hopf trace formula, in the combinatorial context of a partially ordered set. This setting allows us to work with simplicial homology. We begin with the basic deenitions of poset homol-ogy. The Hopf trace theorem is stated and proved. For a group G acting on an

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