Equivariant weak {$n$}-equivalences

Abstract

The notion of n-type was introduced by J.H.C. Whitehead ([22, 23]) where its clear geometric meaning was presented. Following J.L. Hernandez and T. Porter ([12, 13]) we use the term weak n-equivalence for a map f : X → Y of path-connected spaces which induces isomorphisms πk(f) : πk(X)→ πk(Y ) on homotopy groups for k ≤ n. Certainly, weak n-equivalence of a map determines its n-connectedness but not conversely. For J.H. Baues ([2, page 364]) n-types denote the category of spaces X with πk(X) = 0 for k > n. The n-type of a CW -space X is represented by PnX, the n-th term in the Postnikov decomposition of X. Then the n-th Postnikov section pn : X → PnX is a weak n-equivalence. Much work has been done to classify the n-types and find equivalent conditions for a map f : X → Y to be a weak nequivalence. J.L. Hernandez and T. Porter ([12]) showed how with this notion of weak n-equivalence and with a suitable notion of n-fibration and n-cofibration one obtains a Quillen model category structure ([20]) on the category of spaces. The case of weak n-equivalences mod a class C of groups (in the sense of Serre) was analyzed by C. Biasi and the second author ([3]). E. Dror ([5]) pointed out that weak equivalences of certain spaces (including nilpotent and complete spaces) can be described by means of homology groups. Then in 1977 J.H. Baues ([1]) proved the Dual Whitehead Theorem for maps of R-Postnikov spaces (of order k ≥ 1), where R is a commutative ring. Given the growing interest in equivariant homotopy, it is not surprising that notions of equivariant n-types have been studied. For instance algebraic models for equivariant 2-types have been presented by I. Moerdijk and J.-A. Svensson ([18])