Kári Ragnarsson Department of Mathematical Sciences DePaul University 2320 North Kenmore Avenue Chicago, IL 60614 773-325-2061 (phone) 773-325-7807 (fax) Office hours: Tuesdays and Wednesdays 4:30-5:30pm Curriculum vitae (last updated 7/22/2010)

I am a Visiting Assistant Professor of Mathematics at DePaul University and a Research Affiliate of The Mathematics Institute at Reykjavík University. My research interests are in algebraic topology and representation theory of finite groups. I am currently studying p-local properties of classifying spaces of finite groups. This involves fusion systems and p-local finite groups, but also stable homotopy theory of p-completed classifying spaces. Recently I have also worked on some combinatorial projects.

Publications           click arrows next to titles to view/hide abstracts

1. Homology of the Boolean Complex (with Bridget Tenner)
   submitted
   arXiv: 1005.4411

   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 unlabeled Coxeter 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.

2. Completion of G-spectra and stable maps between classifying spaces
   submitted
   arXiv: 1001.0771

   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 G×K. In particular this describes the dual of BG.

3. Saturated fusion systems as idempotents in the double Burnside ring (with R. Stancu)
   submitted
   arXiv: 0911.0085

   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. Applying our results to p-local finite groups, we show that p-local finite groups are equivalent to retractive transfer triples.

4. p-local finite group cohomology (with R. Levi)
   submitted
   arXiv: 0910.5639

   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.

5. Boolean complexes for Ferrers graphs (with A. Claesson, S. Kitaev, and B. E. Tenner)
   submitted
   arXiv: 0808.2307

   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.

6. Obtainable sizes of topologies on finite sets (with B. E. Tenner)
   Journal of Combinatorial Theory, Series A 117 (2010), 138-151
   arXiv: 0802.2550

   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.

7. Homotopy type of the boolean complex of a Coxeter system (with B. E. Tenner)
   Advances in Mathematics 222 (2009), 409-430
   arXiv: 0806.0906

   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, this 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 this 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.

8. Retractive transfers and p-local finite groups
   Proceedings of the Edinburgh Mathematical Society 51 (2008), 465-487
   arXiv: math/0510374

   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, of whether an equivalent theory can be obtained by studying triples (f,t,X), where X is a p-complete, nilpotent space with a finite fundamental group, f:BS → X 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. Using previous results obtained by the author, we show that the converse is true for general finite p-groups. That is, for a p-local finite group (S,\mathcal{F},\mathcal{L}), the natural inclusion θ:BS → X has a retractive transfer t, making (θ,t,|\mathcal{L}|^{\wedge}_p) a retractive transfer triple over S. This also requires a proof, obtained jointly with Ran Levi, that |\mathcal{L}|^{\wedge}_p is a nilpotent space, which is of independent interest.

9. A Segal conjecture for p-completed classifying spaces
   Advances in Mathematics 215 (2007), 540-568
   arXiv: math/0503067

   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 Ap(G,K) 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.

10. Classifying spectra of saturated fusion systems
    Algebraic and Geometric Topology 6 (2006), 195-252
    arXiv: math/0502092

    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.

11. Alternative stable homotopy classification of BGp^{^}
    Topology 45 (2006), 601-609
    arXiv: math/0502577

    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 Σ^∞BGp^{^} as an object under Σ^∞BS 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 homotopy type of BGp^{^} is determined by the map Σ^∞BS → Σ^∞BGp^{^}, but not by the homotopy type of Σ^∞BGp^{^}.

12. Frobenius transfers and p-local finite groups
    Ph.D. Thesis, Massachusetts Institute of Technology, 2004
    

    In this thesis we explore the possibility of defining the p-local finite groups of Broto, Levi and Oliver in terms of their classifying spaces. More precisely, we consider the question posed by Haynes Miller, whether an equivalent theory can be recovered by studying maps f: BS → X from the classifying space of a finite p-group S to a p-complete space X equipped with a stable retract t satisfying a form of Frobenius reciprocity. In the case where S is elementary abelian, we answer this question in the affirmative, by showing that under some finiteness conditions such a triple (f, t, X) does indeed induce a p-local finite group over S. We also discuss the converse in some detail for general S.

Slides           from selected talks and lectures

Encoding fusion data in the double Burnside ring, UIC Algebra Seminar, October 2008 [pdf] Encoding fusion data in the double Burnside ring, MSRI Postdoctoral Seminar, April 2008 [pdf] A Segal conjecture for p-completed classifying spaces, MIT Topology Seminar, November 2005 [pdf] Lecture series on Frobenius transfers, p-local finite groups, and classifying spectra of saturated fusion systems, University of Aberdeen, 2004-2005 [1] [2] [3] [4-5] [6]