Dietrich Notbohm

On Davis–Januszkiewicz homotopy types I ; formality and rationalisation

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose inte- gral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewiczs examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a co- homology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivans commutative cochain algebra, and deduce that the rationali- sation of c(K) is unique for a special family of complexes K. In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivans arithmetic square to produce global results for this family. AMS Classification 55P62, 55U05; 05E99

Maps between classifying spaces and applications

Abstract For connected Lie groups G and H the calculation of the mapping space map( BG , BH ) can be reduced to the case of simply connected Lie groups. This reduction method allows some applications. For example a homotopy classification of self-maps BG → BG which induce rational homotopy equivalences and a classification of fake Lie groups up to homotopy.

Homotopy uniqueness of classifying spaces of compact connected lie groups at primes dividing the order of the weyl group

As A truism, Lie groups-in particular compact connected Lie groups-are very rigid objects. The perhaps best known instance of this rigidity was formulated in Hilbert’s fifth problem and proved by Gleason, Montgomery and Zippin in the early 1950s: It requires only very weak assumptions on the topology of a topological group to get a Lie group (for a survey see [19]). Trying to distinguish two compact connected Lie groups, another kind of rigidity occurs. Very often, the rich structure of a Lie group is totally described by little information. For example, simply connected compact Lie groups or compact connected Lie groups up to the local isomorphism type are classified by pure combinatorical data, namely the Dynkin diagram. Semi simple Lie groups are distinguished by their normalizer of the maximal torus [ 121. Similar phenomena seem to occur, if one considers the classifying space BG of a compact connected Lie group G. Surprisingly, pure algebraic data, given by cohomology or complex K-theory, is enough to distinguish BG as a space from other spaces. Which is what most of the paper is about. p-adic completion of spaces makes life a lot easier. Most of the results are about the p-adic completion BG,^. For a large class of compact connected Lie groups, ‘global’ results are also obtained. We are concerned with three concepts: The homotopy type, the p-adic type, and the mod-p type of a classifying space. We will explain these concepts in detail in a moment. The last two notions are purely algebraic. Each concept is weaker than the preceding one. The main theorems will say that, under certain conditions, the first two are equivalent and characterize the homotopy type of BG,^. That is what we understand by homotopy uniqueness. We will use the following notation throughout: T, 4 G denotes a fixed maximal torus of G,N(T,)csG the normalizer of T,, and W, the Weyl group of G. We say a p-complete space X has the mod-p type of BG, if there exists an isomorphism

Finite loop spaces are manifolds

One of the motivating questions for surgery theory was whether every finite H:space is homotopy equivalent to a Lie group. This question was answered in the negative by Hilton and Roitberg s discovery of some counterexamples [18]. However, the problem remained whether every finite H-space is homotopy equivalent to a closed, smooth manifold. This question is still open, but in case the H-space admits a classifying space we have the following theorem.

Connected finite loop spaces with maximal tori

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says said that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analoguous local problems.

Kernels of maps between classifying spaces

For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that the kernel is a normal subgroup in a modified sense and prove a generalization of a theorem of Quillen, namely, a mapf:BG→BHp∧ is injective, iff the induced map in mod-p cohomology is finite. Moreover, for compact connected Lie groups, every mapf:BG→BHp∧ factors over a quotient ofG in a modified sense and this factorisation is an injection.

p-adic Lattices of Pseudo Reflection Groups

Let U be a vector space over the p-adic rationals, and let W → Gl(U) be faithful representation of a finite group such that W is generated by pseudo reflections. For odd primes we study the p-adic W-lattices of this representation and achieve a complete classification. Examples of such situations are given by the Weyl group acting on the 1-dimesional homology of the maximal torus of a connected compact Lie group, or of the so called p-compact groups, a homotopy theoretic generalisation of compact Lie groups. The associated lattices are an important algebraic invariant in the study of these geometric object.

HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS

For a prime p , a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod- p homology. Associated to a modular representation G → Gl( n ; [ ] p ), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ] p -vector space V , this collection consists of all subgroups of G with nontrivial p -Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG .

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.

A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S3 has the form F2[X2n] ⊗ E(Sq1X2n) where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.