On noncontractible compacta with trivial homology and homotopy groups

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.