Decomposable Free Loop Spaces

Abstract

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

Splitting of gauge groups

“…Then we expect that the loop spaces of H(k, l)-spaces form a new class of higher homotopy commutativity. Kawamoto and Hemmi [12] introduced H k (n)-spaces in order to unify Aguadé's T k -spaces [1] and Félix and Tanré's H(n)-spaces [6]. They also introduced higher homotopy commutativity called C k (n)-spaces in order to describe H k (n)-spaces by higher homotopy.…”

G'p-SPACES FOR MAPS AND HOMOLOGY DECOMPOSITIONS

“…Thus we have, from Proposition 2.1 (5), that H n (A) = i * (H n (X)) = i * (G n (X, r, A)) ⊂ G n (A, ri, A) = G n (A) and A is a G -space. (2) We show that H n (X) ⊂ G n (X) for all n. We obtain, from Proposition 2.1 (1), that H n (X) = G n (X, r, A) ⊂ G n (X, ir, X) = G n (X, 1, X) = G n (X). Thus we know that X is a G -space.…”

Principal Fibrations and Generalized H-Spaces

“…In 1951, Postnikov [13] introduced the notion of the Postnikov system as follows; A Postnikov system for X( or homotopy decomposition of X) {X n , i n , p n } consists of a sequence of spaces and maps satisfying (1) …”

“…The symbols e and e ′ denote τ −1 (1 ΩX )and τ (1 ΣX ) respectively. It is well known [1] that a space X is a T -space if and only if the evaluating map e : ΣΩX → X is cyclic. We called a space X as a T f -space for a map f : A → X [18] if e : ΣΩX → X is f -cyclic, that is, there is a T f -structure F : ΣΩX × A → X such that F j ∼ ∇(e ∨ f ), where j : ΣΩX ∨ A → ΣΩX × A is the inclusion.…”

“…We showed [17] that if a space X is an H-space, then for any space A and any map f : A → X, X is an H f -space for a map f : A → X, but the converse does not hold. In [1], Aguade introduced a T -space as a space X having the property that the evaluation fibration ΩX → X S 1 → X is fibre homotopically trivial. It is easy to show that any H-space is a T -space.…”