On homology of complements of compact sets in Hilbert Cube

TL;DR

This paper investigates the homological properties of complements of compact sets in the Hilbert cube, establishing acyclicity results for certain classes of compacta using weak relative cohomology.

Contribution

It introduces the concept of spaces with weak relative cohomology and proves acyclicity of their complements in the Hilbert cube, extending previous results.

Findings

Complement of weakly infinite dimensional compacta is acyclic.

Complement of compacta with finite cohomological dimension is acyclic.

Introduces weak relative cohomology as a tool for homological analysis.

Abstract

We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of the complement results when (a) $X$ is weakly infinite dimensional; (b) $X$ has finite cohomological dimension.