Fine shape III: $\Delta$-spaces and $\nabla$-spaces

TL;DR

This paper demonstrates that fine shape theory remains manageable for non-locally compact spaces and provides new insights into the structure of infinite-dimensional metrizable spaces using inverse sequences of simplicial complexes.

Contribution

It establishes that every Polish space is fine shape equivalent to an inverse limit of simplicial complexes, with special cases for locally finite dimensional spaces.

Findings

Polish spaces are fine shape equivalent to inverse limits of simplicial complexes.

Non-degenerate simplicial maps can be chosen for locally finite dimensional spaces.

Non-degenerate maps cannot be chosen for Taylor compactum.

Abstract

In this paper we obtain results indicating that fine shape is tractable and "not too strong" even in the non-locally compact case, and can be used to better understand infinite-dimensional metrizable spaces and their homology theories. We show that every Polish space $X$ is fine shape equivalent to the limit of an inverse sequence of simplicial maps between metric simplicial complexes. A deeper result is that if $X$ is locally finite dimensional, then the simplicial maps can be chosen to be non-degenerate. They cannot be chosen to be non-degenerate if $X$ is the Taylor compactum.