Spaces Of Compact Lie Groups Research Articles

Local Gorenstein duality in chromatic group cohomology

We consider local Gorenstein duality for cochain spectra C⁎(BG;R) on the classifying spaces of compact Lie groups G over complex orientable ring spectra R. We show that it holds systematically for a large array of examples of ring spectra R, including Lubin-Tate theories, topological K-theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples.

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=Un,SOn. Extending these results to groups of other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie(K)⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G=KC.We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K-spaces.An essential tool in our study is the dual Poisson-Lie group K⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K⁎, and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie(K)⁎→K⁎.

Geodesic orbit spaces of compact Lie groups of rank two

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H, g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

Homogeneous hypercomplex structures II - Coset spaces of compact Lie groups

In this second in a series paper we complete our classification of hypercomplex manifolds, on which a compact group of automorphisms acts transitively. The description of the spaces as well as the proofs of our results use only the structure theory of reductive groups.

In search of infinite-dimensional Kähler geometry

This paper is devoted to a survey of recent results in the Kähler geometry of infinite-dimensional Kähler manifolds. Three particular classes of such manifolds are investigated: the loop spaces of compact Lie groups, Hilbert–Schmidt Grassmannians, and the universal Teichmüller space. These investigations have been prompted both by requirements in Kähler geometry itself and by connections with string theory, which are considered in the last section. Bibliography: 43 titles.

Borel cohomology and the relative Gorenstein condition for classifying spaces of compact Lie groups

For a compact Lie group G we show that the representing spectrum for Borel cohomology generates its category of modules if G is connected or if coefficients are of characteristic p and π0(G) is a p-group. For a closed subgroup H of G we consider the map C⁎(BG)⟶C⁎(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.

The foundations of -manifolds

The focus of our paper is a system of axioms that serves as a basis for introducing structural data for -manifolds , where is a smooth, compact -dimensional manifold with a smooth effective action of the -dimensional torus . In terms of these data a construction of a model space with an action of the torus is given such that there exists a -equivariant homeomorphism . This homeomorphism induces a homeomorphism . The number is called the complexity of a -manifold. Our theory comprises toric geometry and toric topology, where . It is shown that the class of homogeneous spaces of compact Lie groups, where , contains -manifolds that have nonzero complexity. The results are demonstrated on the complex Grassmann manifolds with an effective action of the torus . Bibliography: 23 titles.

Cobordisms, Manifolds with Torus Action, and Functional Equations

The paper is devoted to applications of functional equations to well-known problems of compact torus actions on oriented smooth manifolds. These include the problem of Hirzebruch genera of complex cobordism classes that are determined by complex, almost complex, and stably complex structures on a fixed manifold. We consider actions with connected stabilizer subgroups. For each such action with isolated fixed points, we introduce rigidity functional equations. This is based on the localization theorem for equivariant Hirzebruch genera. We consider actions of maximal tori on homogeneous spaces of compact Lie groups and torus actions on toric and quasitoric manifolds. The arising class of equations contains both classical and new functional equations that play an important role in modern mathematical physics.

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.

On factor spaces of compact Lie groups by subgroups of corank one

We obtain a classification of transitive actions of connected compact Lie groups on simply connected spaces of the form G/H where H is a subgroup of corank one coinciding with the commutant of the centralizer of a torus.

Fourier coefficients of invariant random fields on homogeneous spaces of compact Lie groups

Soit $T$ un champ aleatoire invariant par rapport a l’action d’un groupe compact $G$. On etudie les proprietes de ses coefficients de Fourier telles que l’orthogonalite et la gaussianite. En particulier on etablit des conditions qui garantissent que l’independance de ces coefficients entraine qu’ils sont gaussiens. Une consequence remarquable est que, en general, il n’est pas possible de generer par simulation un champ aleatoire non gaussien invariant a l’aide de son developpement par des coefficients independants.

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.

The Yang–Mills gradient flow and loop spaces of compact Lie groups

Abstract We study the L 2 ${L^{2}}$ gradient flow of the Yang–Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S 2 ${S^{2}}$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space Ω ⁢ G ${\Omega G}$ of based loops in the compact Lie group G. An isomorphism between these two Morse homologies is obtained by coupling a perturbed version of the Yang–Mills gradient flow with the L 2 ${L^{2}}$ gradient flow of the classical energy functional on loops. Our result gives a positive answer to a question due to Atiyah.

Homotopy idempotent functors on classifying spaces

Fix a prime p p . Since their definition in the context of localization theory, the homotopy functors P B Z / p P_{B\mathbb {Z}/p} and C W B Z / p CW_{B\mathbb {Z}/p} have shown to be powerful tools used to understand and describe the mod p p structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod p p homotopy, such as Bousfield-Kan completion and Bousfield homological localization.

Pairings and monomorphisms of classifying spaces

We consider the maps between classifying spaces of compact Lie groups of the form BK×BL→BG. If the restriction map BL→BG is a weak epimorphism, then the restriction on BK is known to factor through the classifying spaces of the center of the compact Lie group G. Suppose H is a semi-simple subgroup of a connected compact Lie group G with rank(H)=rank(G). Replacing the weak epimorphism BL→BG by the map BH→BG, analogous results are obtained. We also consider some monomorphisms of classifying spaces of compact Lie groups, such as BSO(n)→BSU(n). Our proof will make use of admissible maps.

Infinite dimensional manifolds from a new point of view

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of categories and functors. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined and the basic properties (Functorial Property, Homotopy Invariant, Mayer–Vietoris Sequence) are preserved. In this setting we define the classifying space BG of a Lie group G as an infinite dimensional manifold. Using simplicial homotopy theory and the Chern–Weil theory for principal G-bundles we show that de Rhamʼs theorem holds for BG when G is compact. Finally we get, as an unexpected byproduct, two simplicial set models for the classifying spaces of compact Lie groups; they are totally different from the classical models constructed by Milnor, Milgram, Segal and Steenrod.

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

Geodesic flows on Riemannian g.o. spaces

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.

Noetherian loop spaces

The class of loop spaces of which the mod p cohomology is Noetherian is much larger than the class of p -compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as \mathbb C P^\infty and 3 -connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of B\mathbb C P^\infty . We also show that BX differs basically from the classifying space of a p -compact group in a single homotopy group. This applies in particular to 4 -connected covers of classifying spaces of compact Lie groups and sheds new light on how the cohomology of such an object looks like.

On the cohomology of free and twisted loop spaces

A natural extension of the cohomology suspension to a free loop space is constructed from the evaluation map and is shown to have good properties in cohomology calculation. This map is generalized to a twisted loop space which is a space of paths twisted by a given self-map of the underlying space. As an application, the cohomology of free and twisted loop spaces of classifying spaces of compact Lie groups, including some finite Chevalley groups, is calculated.