Equivariant evaluation on free loop spaces

There is a natural evaluation map on the free loop space which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements . We study the induced map in -equivariant cohomology with mod two coefficients in the cases where for . Received: 17 February 2000; in final form: 28 February 2001 / Published online: 18 January 2002     

Homotopy equivalences of p-completed classifying spaces of finite groups

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ _p ^c (G) with the following properties. Two p-completed classifying spaces BG _p ^∧ and BG’ _p ^∧ have the same homotopy type if and only if the associated categories ℒ _p ^c (G) and ℒ _p ^c (G’) are equivalent. Furthermore, the topological monoid Aut(BG _p ^∧) of self equivalences is determined by the self equivalences of the associated category ℒ _p ^c (G). Oblatum 5-VII-2001 & 28-VIII-2002?Published online: 8 November 2002 RID="*" ID="*"C. Broto is partially supported by DGICYT grant PB97–0203. RID="**" ID="**"R. Levi is partially supported by EPSRC grant GR/M7831. RID="***" ID="***"B. Oliver is partially supported by UMR 7539 of the CNRS.     

Equivariant Reidemeister torsion on symmetric spaces

We calculate explicitly the equivariant Ray-Singer torsion for all symmetric spaces of compact type with respect to the action of . We show that it equals zero except for the odd-dimensional Gra?mannians and the space . As a corollary, we classify up to diffeomorphism all isometries of these spaces which are homotopic to the identity; also, we classify their quotients by finite group actions up to homeomorphism. Received: 31 May 1995 / In revised form: 9 January 1996     

The weak conjecture on spherical classes

Let be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, , to and is the inclusion of the Dickson algebra, , into . This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in }, (2) or a conjecture on ${mathcal{A}}$ -decomposability of the Dickson algebra in $Gamma_k^{wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2). Received November 27, 1996; in final form March 6, 1998     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997     

Homotopy uniqueness of BG 2

We prove that the homotopy type of BG ₂ is determined by its mod 2 cohomology as well as by its Weyl group data. Received: 21 April 1997 / Revised version: 8 December 1997     

Irregularity of certain algebraic fiber spaces

Let X, Y be smooth complex projective varieties, and be a fiber space whose general fiber is a curve of genus g. Denote by q _f the relative irregularity of f. It is proved that , if f is not generically trivial; moreover, if either a) f is non-constant and the general fiber is either hyperelliptic or bielliptic or b) q(Y)= 0, then , and the bound is best possible. A classification of fiber surfaces of genus 3 with q _f = 2 is also given in this note. Received: 19 March 1997 / Revised version: 29 October 1997     

Normalizers of maximal tori

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”. Received April 1, 1995; in final form August 11, 1997     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Local isometries of function spaces

Abstract. We show that for a large class of function spaces any isometry that coincides locally with a surjective isometry must be automatically surjective. This class includes finite-codimensional subspaces of and spaces of E-valued continuous functions for finite-dimensional or uniformly convex and algebraically reflexive E. Received: 5 November 2001 / Published online: 14 February 2003 Thanks: Research of both authors partially supported by a grant \# 1102386 from the NSF and a grant \# DST/INT/US(NSF-RP041)/2000 from the DST