Miller spaces and spherical resolvability of finite complexes

Jeffrey Strom¹

¹ Western Michigan University Kalamazoo, MI 49008, U.S.A.

Fundamenta Mathematicae, Tome 178 (2003) no. 2, pp. 97-108

Résumé

Let ${\mathcal A}$ be a fixed collection of spaces, and suppose $K$ is a nilpotent space that can be built from spaces in ${\mathcal A}$ by a succession of cofiber sequences. We show that, under mild conditions on the collection ${\mathcal A}$, it is possible to construct $K$ from spaces in ${\mathcal A}$ using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if $K$ is a nilpotent finite complex, then ${\mit \Omega }K$ can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied to show that if ${\rm map}_*(X,S^n)$ is weakly contractible for all sufficiently large $n$, then ${\rm map}_*(X,K)$ is weakly contractible for any nilpotent finite complex $K$.

Détail

DOI : 10.4064/fm178-2-1

Keywords: mathcal fixed collection spaces suppose nilpotent space built spaces mathcal succession cofiber sequences under mild conditions collection mathcal possible construct spaces mathcal using instead homotopy inverse limits extensions fibrations consequence nilpotent finite complex mit omega built finite wedges spheres using homotopy limits extensions fibrations applied map * weakly contractible sufficiently large map * weakly contractible nilpotent finite complex