Fiber Strong Shape Theory for Topological Spaces

TL;DR

This paper develops a fiber strong shape theory for topological spaces over a fixed metrizable space, providing a classification that is intermediate in fineness between fiber homotopy and fiber shape theories.

Contribution

It introduces a fiber strong shape theory using fiber preserving analogues of resolutions, extending Mardešić-Lisica's method to classify spaces over a fixed base.

Findings

Provides a classification of spaces over $o$ that is coarser than fiber homotopy but finer than usual fiber shape.

Uses fiber preserving analogues of resolutions instead of traditional resolutions.

Extends existing shape theories to a more refined classification framework.

Abstract

In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space $\Bo$. Our approach is based on the method of Marde\v{s}i\'{c}-Lisica and instead of resolutions, introduced by Marde\v{s}i\'{c}, their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over $\Bo$ which is coarser than the classification of spaces over $\Bo$ induced by fiber homotopy theory, but is finer than the classification of spaces over $\Bo$ given by usual fiber shape theory.