Small loop spaces

The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra in [2] and [3]. A small loop is a loop which is homotopic to a loop contained in an arbitrarily small neighborhood of its base point and a small loop space is a topological space in which every loop is small. Small loops are the strongest obstruction to semi-locally simply connectedness. We construct a small loop space using the Harmonic Archipelago. Furthermore, we define the small loop group of a space and study its impact on covering spaces, in particular its contribution to the fundamental group of the universal covering space.     

Homotopy groups of moduli spaces of representations

We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott-Morse theory on the corresponding moduli spaces.     

On loop spaces of configuration spaces

This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".

     

Homotopy stability of orthogonal spaces in commutative banach algebras

We study the concept of real stable rank for a complex commutative Banach algebra A (rsr A). It is shown that this invariant has a behaviour completely analogous to the classical Bass stable rank; in particular, we establish a precise relation between both invariants.Further, we use this machinery to show that the connected components of the orthogonal spaces of A, O _k ⁿ (A)={ A ^k×n:^t=id_k×k}, stabilize when k and n increase in a way that depends on rsr A Finally, we prove an analogous stabilization result for the homotopy classes of maps from the j-sphere into O _k ⁿ (A), where j is an arbitrary positive integer.Research supported by a grant of the CONICET, Argentina.     

Twistor spaces and spinors over loop spaces

In this paper two natural twistor spaces over the loop space of a Riemannian manifold are constructed and their equivalence is shown in the Kählerian case. This relies on a detailed study of frame bundles of loop spaces on the one hand and, on the other hand, on an explicit local trivialization of the Atiyah operator family [defined in Atiyah (SMF 131:43–59, 1985)] associated to a loop space. We relate these constructions to the Dixmier-Douady obstruction class against the existence of a string structure, as well as to pseudo - line bundle gerbes in the sense of Brylinski (Loop spaces, characteristic classes and geometric quantization. Birkhäuser, Basel, 1993).     

Homotopy monomorphisms and H-splitting in loop space fibrations

Let be a fibration, the holonomy action of this fibration and the connecting map. It is shown that if the fibre F admits an H-structure ν such that ρ?ν○(1×∂) (principal fibrations of all kinds satisfy such a condition), then i is a monomorphism if and only if it is weak monomorphism, the latter is equivalent to that Ωp has a homotopy right inverse Γ. If in addition Γ is an H-map, then ΩE has the same H-type as ΩB×ΩF.     

Homotopy of spaces of diffeomorphisms of some three-dimensional manifolds

The goal of the paper is to calculate the homotopy type of the space of diffeomorphisms for most orientable three-dimensional manifolds with finite fundamental group containing the Klein bottle. The fundamental group of such a manifold Q has the form <a, b ¦abab ^–1=1,a ^mb²ⁿ=1>. As m and n one can have any relatively prime natural numbers; these numbers m, n determine the manifold Q up to diffeomorphism. Let K be a Klein bottle lying in Q and let P be a closed tubular neighborhood in Q of this Klein bottle K. We denote by Diff_o(Q) the connected component of the space of diffeomorphisms QQ containing id Q, and by E₀(K, Q) the connected component of the space of imbeddings KQ containing the inclusion KQ; analogously we define E₀(K, P). The main results of the paper are the following two theorems. THEOREM 1. If m, n1, then the space Diff_o(Q) is homotopy equivalent with a circle. THEOREM 2. If m, n1, then the inclusion E₀(K, P) E₀(K, Q) is a homotopy equivalence. With the help of familiar results on spaces of diffeomorphisms of irreducible manifolds which are sufficiently large, Theorem 1 reduces without difficulty to Theorem 2. The main difficulty is the proof of Theorem 2. This proof develops a technique of Hatcher and the author which deals with spaces of PL-homeomorphisms and diffeomorphisms of irreducible manifolds which are sufficiently large. In the paper we use a different structure definition of the class of manifolds considered. It is easy to verify that these definitions are equivalent.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 72–103, 1982.     

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.