Green equivalences in equivariant mathematics

Equivariant weak n-equivalences

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 bu...

Spaces 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.

Equivariant Homotopy Epimorphisms, Homotopy Monomorphisms and Homotopy Equivalences

Isovariant 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 ...

Equivalences in Bicategories

In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In par...