arXiv subjects

Ran Levi

Publications and source records attributed to Ran Levi.

27 records · Page 2Linked to original sources

arXiv2013

An algebraic model for finite loop spaces

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group.

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arXiv2011

Loop space homology associated to the mod 2 Dickson invariants

The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop space homology of the 2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.

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arXiv2011

Unstable Adams operations on p-local compact groups

A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this paper we define unstable Adams operations within the theory of p-local compact groups, and show that such operations exist under rather mild conditions. More precisely, we prove that for a given p-local compact group G and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of p-adic units which are congruent to 1 modulo p^m to the group of unstable Adams operations on G

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arXiv2011

p-local finite group cohomology

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.

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arXiv2009

The work of Fred Cohen

This paper gives an overview of Fred Cohen's work and is a summary of the talk which I gave during his 60th birthday conference, held at the University of Tokyo in July 2005.

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arXiv2007

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given in (math.AT/0505559) to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

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arXiv2005

Subgroup families controlling p-local finite groups

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F-centric F-radical subgroups (at a minimum) to the set of F-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups.

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arXiv2005

Extension of p-local finite groups

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arXiv2005

Construction of 2-local finite groups of a type studied by Solomon and Benson

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4). Erratum (11 Feb 2005). An error in the paper was pointed out to the authors by Andy Chernak. The error is corrected in the erratum at the end of version 2, which should be read alongside the paper.

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