Postnikov factorizations at infinity

Resumo

We have developed Postnikov sections for Brown-Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces, which is an extension of the proper category. The homotopy fibre of a fibration in the factorization associated with Brown-Grossman groups is an Eilenberg-Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Steenrod homotopy groups. For a space which is first countable at infinity, one of these groups is given by the inverse limit of the homotopy groups of the neighbourhoods at infinity, the other group is isomorphic to the first derived of the inverse limit of this system of groups. In the factorization associated with Steenrod groups the homotopy fibre is an Eilenberg-Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Brown-Grossman homotopy groups. We also obtain a mix factorization containing both kinds of previous factorizations and having homotopy fibres which are Eilenberg-Mac Lane exterior spaces for both kinds of groups. Given a compact metric space embedded in the Hilbert cube, its open neighbourhoods provide the Hilbert cube the structure of an exterior space and the homotopy fibres of the factorizations above are Eilenberg-Mac Lane exterior spaces with respect to inward (or approaching) Quigley groups. © 2004 Elsevier B.V. All rights reserved.