We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.

. We investigate the topological properties of the poset of proper cosets xH in a finite group G. Of particular interest is the reduced Euler characteristic, which is closely related to the value at \Gamma1 of the probabilistic zeta function of G. Our main result gives divisibility properties of this reduced Euler characteristic. 1. Introduction For a finite group G and a non-negative integer s, let P (G; s) be the probability that a randomly chosen ordered s-tuple from G generates G. Philip Hall [16] gave an explicit formula for P (G; s), exhibiting the latter as a finite Dirichlet series P n ann \Gammas , with an 2 Z and an = 0 unless n divides jGj. For example, P (A 5 ; s) = 1 \Gamma 5 5 s \Gamma 6 6 s \Gamma 10 10 s + 20 20 s + 60 30 s \Gamma 60 60 s : In view of Hall's formula, we can speak of P (G; s) for an arbitrary complex number s. The reciprocal of this function of s is sometimes called the zeta function of G; see [5, 19]. The present paper arose fr...

Abstract. A collection C of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either the subgroups in C or their centralizers or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections C of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset C which imply comparison results.

of finite groups

Abstract. This work examines the possible projectivity of 2-modular block parts of non-projective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M12, J2, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co3, Ru, O ′ N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a non-principal block part contains a non-projective summand. The case of O ′ N is additionally found to have a non-projective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters. 1.

this paper, and to illustrate the general theory by examining a number of examples. In general, a necessary but certainly not sufficient condition to be imposed on a representation sequence fR n g is that the structure of its members R n is rather well understood, and that computations in connection with group operations may be performed in an effective way. In dealing with possible generalizations of permutation representations we will have to start from some class R of sequences R 0 ; R 1 ; : : : of finite groups, which is a fairly natural and technically sufficiently controlled "neighborhood" of the sequence fS n g, and it is far from clear from the outset what such a class might be. Also, our answer will have to depend on the precise meaning attached in this context to the word "good", i.e., we will have to exhibit a criterion, applicable to sequences

Abstract not found

Contents 1 Introduction 2 2 G-posets 3 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Implications of the G-action . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The order complex and other constructions . . . . . . . . . . . . . . . . . . . 5 2.4 Canonical homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Orbit posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Homotopy equivalences 9 3.1 The Order Homotopy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Contractible carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Quillen's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Wedge decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Applications 15 4.1 The p-subgroup compl

Homotopy type of partially ordered sets (poset for short) play a crucial role in algebraic topology. In fact, every space is weakly equivalent to a simplicial complex which, of course, can be considered as a poset. Posets also arise in more specific contexts as homological decompositions [10, 6, 16, 20] and subgroups complexes associated to

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: