Jesper Michael Møller

Rational isomorphisms of p-compact groups

A rational isomorphism is a p-compact group homomorphism inducing an isomorphism on rational cohomology. Finite covering homomorphisms and nontrivial endomorphisms of simple p-compact groups are rational isomorphisms. It is shown that rational isomorphisms of p-compact groups restrict to admissible rational isomorphisms of the maximal tori and the classification of rational isomorphisms between connected p-compact groups is reduced to the simply connected case. The paper also contains a triviality criterion asserting that, on any connected p-compact group, only the trivial homomorphism induces the trivial map in rational cohomology. 0. Introduction The notion of a p-compact group was introduced by Dwyer and Wilkerson [9] as a homotopy theoretic candidate for a replacement of compact Lie groups. Subsequent investigation in [18] and [8] strengthened the candidacy in finding that much of the internal structure of compact Lie groups does seem to be present also in pcompact groups. The process of gathering support for p-compact groups continues here where the outlining idea is to translate Baum’s paper [3], describing local isomorphism systems of Lie groups, into the setting of p-compact groups. The rational isomorphisms of the title form the most prominent concept of this paper. However, for the sake of stressing the similarity with Lie groups, I shall in this introduction restrict myself to the the particular rational isomorphisms called finite covering homomorphisms: A finite covering homomorphism between p-compact groups is an epimorphism whose kernel is a finite p-group [Definition 2]. It was shown in [18] that any connected p-compact group X admits a finite covering homomorphism q : Y × S → X where Y is a simply connected p-compact group and S is a p-compact torus; the homomorphism q can even be chosen to be what is here called a special finite covering homomorphism. Locally isomorphic p-compact groups [Definition 3] are characterized by having isomorphic finite covering groups of this kind [Proposition 1.5]. As an example of how these concepts behave as our experience with Lie groups tells us they should, the following theorem — containing elements from Theorem 3.3 and Corollary 3.5 — is an almost mechanical translation of the basic lifting property 1991 Mathematics Subject Classification. 55P35, 55S37.

Rational homotopy of spaces of maps into spheres and complex projective spaces

We investigate the rational homotopy classification problem for the components of some function spaces with Sn or cPn as target space.

Normalizers of maximal tori

Abstract. 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”.

Nilpotent spaces of sections

The space of sections of a fibration is nilpotent provided the base is finite CIK-complex and the fiber is nilpotent. Moreover, localization commutes with the formation of section spaces.

Self-homotopy equivalences of group cohomology spaces

Abstract The group of homotopy classes of self-homotopy equivalences of a space with only two nonvanishing homotopy groups is computed by means of a differential in a Lyndon spectral sequence.

Euler characteristics and Möbius algebras of p-subgroup categories☆

Let G be a finite group and p a prime number. We compute the Euler characteristic in the sense of Leinster for some categories of nonidentity p-subgroups of G. The p-subgroup categories considered include the poset S G , the transporter category T ∗ G , the linking category L G , the Frobenius, or fusion, category F G , and the orbit category O G of all nonidentity p-subgroups of G.

Equivariant embeddings of principal Z-bundles into complex vector bundles

Abstract Let π: E→X be a principal Z n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eeE and ze Z n ⊂S 1 ⊂ Z . We compute the primary obstruction ceH 2m (X; Z ) to embedding π equivariantly into p. If dim X⩽2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle em. We use these criteria for a discussion of the classification of the equivalence classes of principal Z -bundles that admit equivariant embeddings into em. Finally, we offer an example of a principal Z -bundle that admit an ordinary but not an equivariant embedding into e1.

Equivariant Euler characteristics of partition posets

We compute all the equivariant Euler characteristics of the

Chromatic Polynomials of Simplicial Complexes

\Sigma_n

On the mod 2 cohomology of spaces of sphere and projective bundle sections

-poset of partitions of the