P-compact Groups Research Articles

Convergent sequences in iterated ultrapowers as p-compact groups

We prove, in ZFC, that if G is an infinite countable Abelian group, then there is an ultrafilter p∈ω⁎ such that (Ultp(G),τBohr‾) has non-trivial convergent sequences, consequently (Ultpω1(G),τBohr‾) has non-trivial convergent sequences, extending Theorem 3.9 from [13]. In addition, we prove that the Remark 3.8 from [13] is false; so, the proof of the Corollary 3.11 is false too.

Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group

This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac–Moody group and source the classifying space of either a p-compact group or a connected Kac–Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac–Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac–Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations. These partial results show that null maps are detected by restricting to the maximal torus in many cases of interest.

A 2-Compact Group as a Spets

In 1993, Broué, Malle and Michel initiated the study of spetses on the Greek island bearing the same name. These are mysterious objects attached to non-real Weyl groups. In algebraic topology, a p-compact group X is a space which is a homotopy-theoretic p-local analogue of a compact Lie group. A connected p-compact group X is determined by its root datum which in turn determines its Weyl group . In this article, we give strong numerical evidence for a connection between these two objects by considering the case when X is the exotic 2-compact group constructed by Dwyer–Wilkerson and is the complex reflection group . Inspired by results in Deligne–Lusztig theory for classical groups, if q is an odd prime power, then we propose a set of “ordinary irreducible characters” associated to the space of homotopy fixed points under the unstable Adams operation ψq. Notably, includes the set of unipotent characters associated to G24 constructed by Broué, Malle and Michel from the Hecke algebra of G24 using the theory of spetses. By regarding as the classifying space of a Benson–Solomon fusion system we formulate and prove an analogue of Robinson’s ordinary weight conjecture that the number of characters of defect d in can be counted locally.

On countably compact group topologies without non-trivial convergent sequences on [formula omitted] for arbitrarily large κ and a selective ultrafilter

Castro Pereira and Tomita showed that it is consistent that there exist arbitrarily large countably compact groups without non-trivial convergent sequences of finite order. In this paper, we obtain large p-compact groups without non-trivial convergent on the direct sum of κ copies of Q where κ=κω is infinite. In particular this gives the first arbitrarily large countably compact groups without non-trivial convergent sequences that are torsion-free.

Selectively pseudocompact groups and p-compactness

We say that a space X is selectively pseudocompact if for each sequence (Un)n<ω of nonempty open subsets of X there is a sequence (xn)n<ω of points in X such that xn∈Un, for each n<ω, and the set {xn:n<ω} has a cluster point in X. We prove that if p and q are not equivalent selective ultrafilters on ω, then there are a p-compact group and a q-compact group whose product is not selectively pseudocompact.

Local Gorenstein duality for cochains on spaces

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of k-algebras. Our main examples are of the form R=C⁎(X;k), the ring spectrum of cochains on a space X for a field k. In particular, we establish local Gorenstein duality in characteristic p for p-compact groups and p-local finite groups as well as for k=Q and X a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.

Stratification and duality for homotopical groups

We generalize Quillen's F-isomorphism theorem, Quillen's stratification theorem, the stable transfer, and the finite generation of cohomology rings from finite groups to homotopical groups. As a consequence, we show that the category of module spectra over C⁎(BG,Fp) is stratified and costratified for a large class of p-local compact groups G including compact Lie groups, connected p-compact groups, and p-local finite groups, thereby giving a support-theoretic classification of all localizing and colocalizing subcategories of this category. Moreover, we prove that p-compact groups admit a homotopical form of Gorenstein duality.

Some new examples of simple $p$-local compact groups

In this paper we present new examples of simple p-local compact groupsfor all odd primes. We also develop the necessary tools to show saturation, simpleness, and the non-realizability as p-compact groups or compact Lie groups, which can be applied in a more general framework.

The dual polyhedral product, cocategory and nilpotence

The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik–Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of ΩX. This answers a fifty year old problem posed by Ganea. The methods are applied to determine the homotopy nilpotency class of quasi-p-regular exceptional Lie groups and sporadic p-compact groups.

Groups of unstable Adams operations on p–local compact groups

A $p$-local compact group is an algebraic object modelled on the homotopy theory associated with $p$-completed classifying spaces of compact Lie groups and p-compact groups. In particular $p$-local compact groups give a unified framework in which one may study $p$-completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p-compact groups, $p$-local compact groups admit unstable Adams operations - self equivalences that are characterised by their cohomological effect. Unstable Adams operations on $p$-local compact groups were constructed in a previous paper by F. Junod and the authors. In the current paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of operations.

GKM theory for p-compact groups

This work studies the flag varieties of p-compact groups, principally through torus-equivariant cohomology, extending methods and tools of classical Schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups. In particular we give, for the infinite family of p-compact flag varieties corresponding to the complex reflection groups G(r,1,n), a generalized GKM characterization (following Goresky–Kottwitz–MacPherson [8]) of the torus-equivariant cohomology, building an explicit additive basis and showing its relationship with the polynomial or Borel presentation via the localization map.

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.

Spaces with Noetherian cohomology

AbstractIs the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

Unstable Adams operations onp–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

The fundamental group of a p -compact group

The notion of p-compact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of p-compact groups, one for each prime number p. A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form. There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus [1, 5.47]. For some time there has in fact been a corresponding formula for the center of a p-compact group [11, 7.5], but in general the fundamental group has eluded analysis. The purpose of the present paper is to remedy this deficit. For any space Y , we let H Zp i (Y ) denotes lim nHi(Y ;Z/p ). Suppose that X is a connected p-compact group, with maximal torus T and torus normalizer NT [10, §8]. It is known that the map π1(T ) → π1(X) is surjective [12, 6.11] [21, 5.6], or equivalently that the map H Zp 2 (BT ) → H Zp 2 (BX) is surjective. We prove the following statement.

$p$-compact groups as subgroups of maximal rank of Kac-Moody groups

In [28], Kitchloo constructed a map f : BX → BK∧ p where K is a certain KacMoody group of rank two, X is a rank two mod p finite loop space and f is such that it induces an isomorphism between even dimensional mod p cohomology groups. Here B denotes the classifying space functor and (−)p denotes the Bousfield-Kan Fp-completion functor ([8]). This space X —or rather the triple (X∧ p , BX ∧ p , e) where e : X ' ΩBX— is a particular example of what is known as a p-compact group. These objects were introduced by Dwyer and Wilkerson in [15] as the homotopy theoretical framework to study finite loop spaces and compact Lie groups from a homotopy point of view. The foundational paper [15] together with its many sequels by Dwyer-Wilkerson and other authors represent now an active, well established research area which contains some of the most important recent advances in homotopy theory. While p-compact groups are nowadays reasonably well understood objects, our understanding of Kac-Moody groups and their classifying spaces from a homotopy point of view is far from satisfactory. The work of Kitchloo in [28] started a project which has also involved Broto, Saumell, Ruiz and the present author and has produced a series of results ([2], [3], [10]) which show interesting similarities between this theory and the theory of p-compact groups, as well as non trivial challenging differences. The goal of this paper is to extend the construction of Kitchloo that we have recalled above to produce rank-preserving maps BX → BK∧ p for a wide family of p-compact groups X. These maps can be understood as the homotopy analogues to monomorphisms, in a sense that will be made precise in section 13. We prove:

The Steenrod problem of realizing polynomial cohomology rings

In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H^*(CP^\infty;Z) = Z[x_2], H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) = Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends Notbohm's solution for R = F_p, p odd. Odd degree generators, excluded above, only occur if R is an F_2-algebra and in that case the recent classification of 2-compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of p-compact groups, but not on classification results for these.

Homotopy type and [formula omitted]-periodic homotopy groups of p-compact groups

We determine the v 1 -periodic homotopy groups of all irreducible p-compact groups ( B X , X ) . In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.

Automorphisms ofp–compact groups and their root data

We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces.

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.