maps between classifying spaces of kac moody groups abstract we classify up to homotopy the self maps of the classifying space of any non affine kac moody group of rank two

classification of problematic subgroups of u n we classify p toral subgroups of u n that can have non contractible fixed points under the action of u n on the com plex ln of partitions of c n into mutually orthogonal subspaces

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

normalizers of maximal tori normalizers and p normalizers of maximal tori in p compact groups can be characterized by the euler characteristic of the associated homogeneous spaces applied to centralizers of elementary abelian p groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus i e that normalizer commutes with centralizer

unramified cohomology of degree 3 and noether s problem let g be a finite group and w be a faithful representation of g over c the group g acts on the field of rational functions c w the question whether the field of invariant functions c w g is purely transcendental over c goes back to emmy noether using the unramified cohomology group of degree 2 of this field as an invariant saltman gave the first examples for which c w g is not rational over c around 1986 bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group g

quillen grassmannians as non modular homotopy fixed points the aim of this paper is to compute the homotopy fixed points of the homotopy action described by unstable adams operations on the classifying spaces of unitary groups the same technique can be applied to compute homotopy fixed points of the action of certain automorphisms on p compact groups called generalized grassmannians we use the description of quillen grassmannians to describe the set of homotopy representations of elementary abelian p groups into them

standard stratifications of ei categories and alperin s weight conjecture abstract we characterize the finite ei categories whose representations are standardly stratified with respect to the natural preorder on the simple representations the orbit category of a finite group with respect to any set of subgroups is always such a category taking the subgroups to be the p subgroups of the group we reformulate alperin s weight conjecture in terms of the standard and proper costandard representations of the orbit category we do this using the properties of the ringel dual construction and a theorem of dlab which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules we indicate that these results hold in the generality of an algebra whose simple modules are preordered rather than partially ordered

existence and uniqueness of linking systems chermak s proof via obstruction theory we present a version of a proof by andy chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems this proof differs from the one in ch2 in that it is based on the computation of derived functors of certain inverse limits this leads to a much shorter proof but one which is aimed mostly at researchers familiar with homological algebra

degree three unramified cohomology groups abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed subfield noether s problem asks whether k g is rational purely transcendental over k the unramified brauer group br nr c g and the unramified cohomology h nr 3 c g q z are obstructions to the rationality of c g see 14 and 5 peyre proves that if p is an odd prime number then there is a group g such that g p 12 br nr c g 0 but h nr 3 c g q z 0 thus c g is not stably c rational 12 using peyre s method we are able to find groups g with g p 9 where p is an odd prime number such that br nr c g 0 h nr 3 c g q z 0

for which pseudo reflection groups are thep adic polynomial invariants again a polynomial algebra abstract letwbe a finite group acting on a latticelover thep adic integers z p the analysis of the ring of invariants of the associatedw action on the algebra z p l of polynomial functions onlis a classical question of invariant theory ifpis coprime to the order ofw classical results show thatwis a pseudo reflection group if and only if the ring of invariants is again polynomial we analyze the situation for odd primes dividing the order ofwand in particular determine those pseudo reflection groups for which the ring of invariants z p l wis a polynomial algebra

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

vector bundles over classifying spaces of compact lie groups the completion theorem of atiyah and segal as says that the complex k theory group k bg of the classifying space of any compact lie group g is isomorphic to r g the representation ring completed with respect to its augmentation ideal however the group k bg bg z bu does not directly contain information about vector bundles over the infinite dimensional complex bg itself the purpose of this paper is to compare the grothendieck group of vector bundles over bg which we denote k bg with both k bg and r g the main result is an algebraic description of k bg in terms of the representation rings of certain subgroups of g as one consequence we show that of the natural maps

fibrations of classifying spaces we investigate fibrations of the form z y x where two of the three spaces are classifying spaces of compact connected lie groups we obtain certain finiteness conditions on the third space which make it also a classifying space our results allow to express some of the basic notions in group theory in terms of homotopy theory i e in terms of classifying spaces as an application we prove that every retract of the classifying space of a compact connected lie group is again a classifying space

equivariant stable homotopy and sullivan s conjecture let g be a p group and let x be a g complex let eg denote a contractible space on which g acts freely by the homotopy fixed point set of x we mean the fixed point set f eg x g where f eg x denotes the function space of all maps eg x equipped with a g action by 9f 9f9 1 if we let denote the one point space with trivial g action we may also consider the g space f x it is canonically g homeomorphic to x the g map e g induces a map x g f x g in 19 d sullivan proposed the following

rationality of p group actions abstract let p be a prime number g a p group of order p 4 k a field with char k p if the exponent of g is p e and k contains a primitive p e th root of unity then k v g is rational purely transcendental over k where g gl v is any linear representation of g over k

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

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 mathcal 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 mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal 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

loop structures on homotopy fibers of self maps of spheres let s 2 n 1 k denote the fiber of the degree k map on the sphere s 2 n 1 if k p r where p is an odd prime and n divides p 1 then s 2 n 1 k is known to be a loop space it is also known that s 3 2 r is a loop space for r 3 in this paper we study the possible loop structures on this family of spaces for all primes p in particular we show that s 3 4 is not a loop space our main result is that whenever s 2 n 1 p r is a loop space the loop structure is unique up to homotopy

rational equivalences between classifying spaces the paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected lie groups with an application to genus sets of such spaces

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

a version of alperinʼs weight conjecture for finite category algebras let p be a prime and k an algebraically closed field of characteristic p we construct a functor c oc on the category of finite categories with the property that if g c is a finite group then oc is the orbit category of p subgroups of g this leads to an extension of alperin s weight conjecture to any finite category c stating that the number of isomorphism classes of simple kc modules should be equal to that of the weight algebra w koc of oc we show that the versions of alperin s weight conjecture for finite groups and for finite categories are in fact equivalent 2013 elsevier inc

representations of categories and their applications we define for each small category c a category algebra rc over a base ring r and study its representations when c is an ei category we develop a theory of vertices and sources for rc mod which parameterizes the indecomposable rc modules as a main application we use our theory to find formulas for computing higher inverse limits over c

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

on the mod p cohomology of bpu p we study the mod p cohomology of the classifying space of the projective unitary group pu p we first prove that conjectures due to j f adams and kono and yagita 1993 about the structure of the mod p cohomology of the classifying space of connected compact lie groups hold in the case of pu p finally we prove that the classifying space of the projective unitary group pu p is determined by its mod p cohomology as an unstable algebra over the steenrod algebra for p 3 completing previous work by dwyer miller and wilkerson 1992 and broto and viruel 1998 for the cases p 2 3

minami webb splittings and some exotic p local finite groups we give a new proof of the minami webb formula for classifying spaces of finite groups by exploiting symonds s resolution of webb s conjecture the methods are applicable to obtain a stable decomposition of minami s type for the classifying spaces of the three exotic p local finite groups which were introduced by ruiz and viruel at the prime 7 we obtain a similar decomposition for the classifying spaces of a family of exotic p local finite groups which were constructed by broto levi and oliver

homology decompositions for p compact groups we construct a homotopy theoretic setup for homology decompositions of classifying spaces of p compact groups this setup is then used to obtain a subgroup decomposition for p compact groups which generalizes the subgroup decomposition with respect to p stubborn subgroups for a compact lie group constructed by jackowski mcclure and oliver

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

group actions on spheres with rank one isotropy let g be a rank two finite group and let h denote the family of all rank one p subgroups of g for which rankp g 2 we show that a rank two finite group g which satisfies certain explicit group theoretic conditions admits a finite g cw complex x s n with isotropy in h whose fixed sets are homotopy spheres our construction provides an infinite family of new non linear g cw complex examples