The K-theory of p-compact groups

In this paper, we show that the p-adic K-theory of a connected p-compact is the ring of invariants of the Weyl group action on the K-theory of a maximal torus. We apply this result to show that a connected finite loop space admits a maximal torus if and only if its complex K-theory is -isomorphic to the K-theory of some BG, where G is a compact connected Lie group. Received: November 9, 1996     

A finite loop space not rationally equivalent to a compact Lie group

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5. Mathematics Subject Classification (2000) Primary: 55P35; Secondary: 55P15, 55R35     

Über den Symmetriegrad der rational azyklischen kompakten homogenen Räume

Actions of compact Lie groups on the homogeneous spaces G/NT, G a compact connected semisimple Lie group, NTG the normalizor of a maximal torus T in G, are considered. If the acting group is a torus, the action is lifted to the universal covering G/T and the corresponding equivariant cohomology is computed for a coefficient field of characteristic O. The symmetry degree of all homogeneous spaces G/NT is computed confirming a conjecture of W. Y. Hsiang. The nonexistence of fixed points of semisimple compact Lie group actions on G/NT is proved in the case that the group acts differentiably and effectively.     

Adjoint action of a finite loop space

Adjoint actions of compact simply connected Lie groups are studied by A. Kono and K. Kozima based on the series of studies on the classification of compact Lie groups and their cohomologies. At odd primes, there is a simpler homotopy theoretic approach that will prove the results of Kono and Kozima for any finite loop spaces. However, there are some technical difficulties at the prime 2.

     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

Distinguished orbits and the L–S category of simply connected compact Lie groups

We show that the Lusternik–Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.     

Maximal antipodal sets of compact classical symmetric spaces and their cardinalities I

We classify and explicitly describe maximal antipodal sets of some compact classical symmetric spaces and those of their quotient spaces by making use of suitable embeddings of these symmetric spaces into compact classical Lie groups. We give the cardinalities of maximal antipodal sets and we determine the maximum of the cardinalities and maximal antipodal sets whose cardinalities attain the maximum.     

A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups

We consider finite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We show that whenever a group covers a Klein bottle weak solenoidal space it covers groups as well, moreover it covers the product of two solenoids. The converse is not true, we give an example of group which covers groups with any finite number of sheets, but does not cover any Klein bottle weak solenoidal space.     

The Holomorphic 2-Number of a Hermitian Symmetric Space

This paper contains a proof a priori (i.e. independent of the classification of Hermitian symmetric spaces) of a theorem on the holomorphic 2-number of a Hermitian symmetric space. If N=G/K is a Hermitian symmetric space, where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,N) = E₁,... , E_m is the fixed point set of T in N, then for each pair E_i, E_j there is a two-dimensional sphere N_ij N such that E_i and E_j are antipodal points of N_ij.     

On the cohomology of homogeneous spaces of finite loop spaces and the Eilenberg–Moore spectral sequence

We prove a collapse theorem for the Eilenberg–Moore spectral sequence and as an application we show that under certain conditions the cohomology of a homogeneous space of a connected finite loop space with a maximal rank torsion free subgroup is concentrated in even degrees and torsionfree, generalizing classical theorems for compact Lie groups of Borel and Bott.     

The finiteness obstruction for loop spaces

For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite H-spaces, that are H-spaces whose homology looks like the homology of a finite CW-complex. In this paper we prove this conjecture for loop spaces. In particular, this shows that every quasi finite loop space is actually homotopy equivalent to a finite CW-complex. Received: March 25, 1999.     

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Compact lie group actions with isotropy subgroups of maximal rank

When a compact connected Lie group acts smoothly on a manifold X with only connected isotropy subgroups of maximal rank, the action is completeley determined by the corresponding action of its Weylgroup WG on the fixed space X^T of the maximal torus. Isotropy subgroups of such actions are determined.     

Suspension Splittings and Self-maps of Flag Manifolds

If G is a compact connected Lie group and T is a maximal torus, we give a wedge decomposition of ΣG/T by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of G/T.     

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.     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

On the homotopy type of the classifying space of the exceptional Lie group F4

 Previous work of several authors shows that the exceptional Lie group of rank 4, F ₄, as a p-compact group, is determined up to isomorphism by the isomorphism type of its maximal torus normalizer for p > 2. This paper considers the case p= 2 proving that F ₄ as 2-compact group is also determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows the authors to determine the integral homotopy type of F ₄ among connected finite loop spaces with maximal tori. Received: 21 June 2000 / Revised version: 4 December 2001