Simplicial Resolutions and Ganea Fibrations

In this work, we compare the two approximations of a pathconnected space X, by the Ganea spaces Gn(X) and by the realizations ‖Λ•X‖n of the truncated simplicial resolutions emerging from the loop-suspension cotriple ΣΩ. For a simply connected space X, we construct maps ‖Λ•X‖n−1 → Gn(X) → ‖Λ•X‖n over X, up to homotopy. In the case n = 2, we prove the existence of a map G2(X) → ‖Λ•X‖1 over X (up to homotopy) and conjecture that this map exists for any n. We use the category Top of well pointed compactly generated spaces having the homotopy type of CW-complexes. We denote by Ω and Σ the classical loop space and (reduced) suspension constructions on Top. Let X ∈ Top. First we recall the construction of the Ganea fibrations Gn(X) → X where Gn(X) has the same homotopy type as the n-th stage, BnΩX , of the construction of the classifying space of ΩX : (1) the first Ganea fibration, p1 : G1(X) → X , is the associated fibration to the evaluation map evX : ΣΩX → X ; (2) given the nth-fibration pn : Gn(X) → X , let Fn(X) be its homotopy fiber and let Gn(X)∪ C(Fn(X)) be the mapping cone of the inclusion Fn(X) → Gn(X). We define now a map p ′ n+1 : Gn(X) ∪ C(Fn(X)) → X as pn on Gn(X) and that sends the (reduced) cone C(Fn(X)) on the base point. The (n + 1)-st-fibration of Ganea, pn+1 : Gn+1(X) → X , is the fibration associated to pn+1. (3) Denote byG∞(X) the direct limit of the canonical mapsGn(X) → Gn+1(X) and by p∞ : G∞(X) → X the map induced by the pn’s. From a classical theorem of Ganea [3], one knows that the fiber of pn has the homotopy type of an (n+1)-fold reduced join of ΩX with itself. Therefore the maps pn are higher and higher connected when the integer n grows. As a consequence, if X is path-connected, the map p∞ : G∞(X) → X is a homotopy equivalence and the total spaces Gn(X) constitute approximations of the space X . The previous construction starts with the couple of adjoint functors Ω and Σ. From them, we can construct a simplicial space Λ•X , defined by ΛnX = (ΣΩ) X and augmented by d0 = evX : ΣΩX → X . Forgetting the degeneracies, we have a facial space (also called restricted simplicial space in [2, 3.13]). Denote by ‖Λ•X‖ the realization of this facial space (see [7] or Section 1). An adaptation of the proof of Stover (see [8, Proposition 3.5]) shows that the augmentation d0 induces a map ‖Λ•X‖ → X which is a homotopy equivalence. If we consider the successive stages of the realization of the facial space Λ•X , we get maps ‖Λ•X‖n → X which constitute a second sequence of approximations of the space X . In this work, we study the relationship between these two sequences of approximations and prove the following results. Date: February 2, 2008. 1 2 T. KAHL, H. SCHEERER, D. TANRÉ, AND L. VANDEMBROUCQ Theorem 1. Let X ∈ Top be a simply connected space. Then there is a homotopy commutative diagram ‖Λ•X‖n−1 // ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Gn(X) //