Postnikov factorizations at infinity (Q2577129)

The category of exterior spaces was introduced in [\textit{J. M. García-Calcines, M. García-Pinillos} and \textit{L. J. Hernández-Paricio}, Bull. Austral. Math. Soc. 57, No. 2, 221--242 (1998; Zbl 0907.55017)] as a useful enveloping structure for the study of proper homotopy theory. An exterior space \((X,\varepsilon \subseteq \tau)\) consists of a topological space \((X,\tau)\) together with a non-empty collection \(\varepsilon\) of open subsets that is upward closed for the inclusion ordering on \(\tau\) and also closed under binary intersection (these exterior open sets are thought of as being `neighbourhoods of infinity'). Extending definitions from proper homotopy theory, in [loc. cit.] the authors introduced exterior analogues of the Brown-Grossman homotopy groups and the Steenrod homotopy groups. These are linked by a long exact sequence (these earlier results are summarised in this paper). In this paper, the authors prove, in general, the existence of Postnikov-type decompositions relative to the Brown-Grossman homotopy groups, and similarly, for a more restricted class of exterior spaces (first countable at infinity plus a connectivity condition `at infinity') relative to the Steenrod homotopy groups. They also obtain a mixed factorisation combining both of the previous ones. Some applications, examples and open questions are discussed in the last section.