Homotopy coherent diagrams and approximate fibrations

TL;DR

This paper demonstrates that certain fibrations over finite simplicial complexes can be approximated by fibrations with ENR total spaces, using homotopy coherent diagrams, with implications for Hilbert cube manifolds.

Contribution

It introduces a method to approximate fibrations with ENR total spaces via homotopy coherent diagrams, extending the understanding of fibering in topological manifolds.

Findings

Fibrations over finite simplicial complexes can be approximated by approximate fibrations with ENR total spaces.

Homotopy coherent diagrams and their homotopy colimits are effective tools in this approximation.

Application to fibering of Hilbert cube manifolds shows practical relevance.

Abstract

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy type of the total space and give an application to the fibering of Hilbert cube manifolds.