Homotopy equivalences in equivariant topology

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 ...

Triangulating and Smoothing Homotopy Equivalences and Homeomorphisms . Geometric Topology Seminar

We will study the smooth and piecewise linear manifolds within a given homotopy equivalence class. In the first part we find an obstruction theory for deforming a homotopy equivalence between manifolds to a diffeomorphism or a piecewise linear homeomorphism. In the second part we analyze the piecewise linear case and characterize the obstructions in terms of a geometric property of the homotopy...

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...

Coglueing Homotopy Equivalences

in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the ba...