On noncontractible compacta with trivial homology and homotopy groups

TL;DR

This paper constructs a special Peano continuum that is weakly homotopy equivalent to a point but noncontractible, with trivial classical homology and cohomology groups, challenging traditional assumptions in topology.

Contribution

It provides a novel example of a noncontractible space with trivial homology and homotopy groups, expanding understanding of topological space properties.

Findings

Constructed a Peano continuum with trivial homotopy groups but noncontractible

Proved all classical homology and cohomology groups of the space are trivial

Showed the space is homologically and cohomologically locally connected

Abstract

We construct an example of a Peano continuum $X$ such that: (i) $X$ is a one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy equivalent to a point (i.e. $\pi_n(X)$ is trivial for all $n \geq 0$); (iii) $X$ is noncontractible; and (iv) $X$ is homologically and cohomologically locally connected (i.e. $X$ is a $HLC$ and $clc$ space). We also prove that all classical homology groups (singular, \v{C}ech, and Borel-Moore), all classical cohomology groups (singular and \v{C}ech), and all finite-dimensional Hawaiian groups of $X$ are trivial.