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In previous work, we associated to any finite simple graph a particular set of derangements of its vertices. These derangements are in bijection with the spheres in the wedge sum describing the homotopy type of the boolean complex for this graph. Here we study the frequency with which a given derangement appears in this set., Comment: 6 pages

We say that a fusion system is the composition product of two subsystems if every morphism can be factored as a morphism in one fusion system followed by a morphism in the other. We establish a relationship between the characteristic idempotent of a saturated fusion system that is the composition product of saturated subsystems and the characteristic idempotents of the component systems. Consequently we obtain a compatibility result for transfer through the composition product and transfer through the component systems., Comment: 8 pages

We construct and analyze an explicit basis for the homology of the boolean complex of a Coxeter system. This gives combinatorial meaning to the spheres in the wedge sum describing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collection of these elements forms a basis for the homology of the boolean complex. In this manner, the spheres in the wedge sum describing the homotopy type of the complex can be represented by a set of derangements. We give an explicit, closed-form description of the derangements that can be obtained from any graph, and compute this set for several families of graphs. In the cases of complete graphs and Ferrers graphs, these calculations give bijective proofs of previously obtained enumerative results., Comment: final version, to appear in Journal of Algebraic Combinatorics

We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime. As an application, we show that the spectrum of stable maps from BG to the classifying space of a compact Lie group K splits non-equivariantly as a wedge sum of p-completed suspension spectra of classifying spaces of certain subquotients of the product of G and K. In particular this describes the dual of BG., Comment: Final version, to appear in Advances in Mathematics

We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Haynes Miller which proposes retractive transfer triples as a purely homotopy-theoretic model for p-local finite groups. We take an important step toward proving this conjecture by showing that a retractive transfer triple gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions., Comment: Added citation to Puig; clarified discussion of Miller's conjecture on the homotopy characterization of p-local finite groups

We study cohomology for $p$-local finite groups with non-constant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences., Comment: 32 pages. Final version, to appear in Homology, Homotopy and Applications

In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legendre-Stirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which is expressed in terms of the Stirling numbers of the second kind. Finally, we analyze the complexity of calculating the boolean number of a Ferrers graph using these results and show that it is a significant improvement over calculating by edge recursion., Comment: final version, to appear in the The Australasian Journal of Combinatorics

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group., Comment: final version, to appear in Advances in Mathematics

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies., Comment: Final version, to appear in Journal of Combinatorial Theory, Series A

In this paper we explore the possibility of defining p-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be obtained by studying triples (f,t,X), where X is a p-complete, nilpotent space with finite fundamental group, f is a map from the classifying space of a finite p-group, and t is a stable retraction of f satisfying Frobenius reciprocity at the level of stable homotopy. We refer to t as a retractive transfer of f and to (f,t,X) as a retractive transfer triple over S. In the case where S is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple (f,t,X) over S does indeed induce a p-local finite group over S with X as its classifying space. Conversely, we show that for a p-local finite group over any finite p-group S, the natural inclusion of BS into the classifying space of the p-local finite group has a retractive transfer t, making the triple consisting of the inclusion, the retractive transfer and the classifying space of the p-local finite group a retractive transfer triple over S. This also requires a proof, obtained jointly with Ran Levi, that the classifying space of a p-local finite group is a nilpotent space, which is of independent interest., Comment: 23 pages, to appear in Proceedings of the Edinburgh Mathematical Society. Replaced Sections 4 and 5 with a single section using Lannes theory to construct a homotopy equivalence of classifying spaces rather than Wojtkowiak's obstruction theory to prove existence of the equivalence. Other minor changes

We formulate and prove a new variant of the Segal Conjecture describing the group of homotopy classes of stable maps from the p-completed classifying space of a finite group G to the classifying space of a compact Lie group K as the p-adic completion of the Grothendieck group of finite principal (G,K)-bundles whose isotropy groups are p-groups. Collecting the result for different primes p, we get a new and simple description of the group of homotopy classes of stable maps between (uncompleted) classifying spaces of groups. This description allows us to determine the kernel of the map from the Grothendieck group A(G,K) of finite principal (G,K)-bundles to the group of homotopy classes of stable maps from BG to BK., Comment: 31 pages. Minor changes. Final version, to appear in Advances in Mathematics

We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space of G as an object under the stable p-completed classifying space of S via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino-Priddy conjecture, we obtain the surprising result that the unstable p-completed homotopy type of BG is determined by the stable p-completed inclusion of BS in BG, but not by the stable p-completed homotopy type of BG., Comment: 10 pages. Clarified text after Definition 1.1 and altered discussion in Section 4 following suggestions from referee. To appear in Topology

The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p-completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying p-group., Comment: This is the version published by Algebraic & Geometric Topology on 26 February 2006