Equivariant weak n-equivalences
نویسندگان
چکیده
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])
منابع مشابه
Elmendorf’s Theorem via Model Categories
In [2], working in the category of compactly generated spaces U , Elmendorf relates the equivariant homotopy theory of G-spaces to a homotopy theory of diagrams using fixed point sets. The diagrams are indexed by a topological category OG with objects the orbit spaces {G/H}H for the closed subgroups H ⊂ G. Although, his general assumption there is that G is a compact Lie group, a formulation of...
متن کاملSpaces of Equivariant Self-equivalences of Spheres
Let F(ST) denote the identity component of the space of homotopy self-equivalences of STM and let F = inj limw FiS" ). This paper studies the homotopy properties of certain equivariant analogs of the infinite loop space F.
متن کاملHomotopy Theories for Diagrams of Spaces
We show that the category of diagrams of topological spaces (or simplicial sets) admits many interesting model category structures in the sense of Quillen [8]. The strongest one renders any diagram of simplicial complexes and simplicial maps between them both fibrant and cofibrant. Namely, homotopy invertible maps between such are the weak equivalences and they are detectable by the "spaces of ...
متن کاملIsovariant extensors and the characterization of equivariant homotopy equivalences
We extend the well-known theorem of James–Segal to the case of an arbitrary family F of conjugacy classes of closed subgroups of a compact Lie group G: a G-map f : X → Y of metric Equiv F -ANE-spaces is a G-homotopy equivalence if and only if it is a weak G-F-homotopy equivalence. The proof is based on the theory of isovariant extensors, which is developed in this paper and enables us to endow ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2000