On noncontractible compacta with trivial homology and homotopy groups

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. π n (X) is trivial for all n > 0); (iii) X is noncontractible; and (iv) X is homologically and cohomologically locally connected (i.e. X is an HLC and clc space). We also prove that all classical homology groups (singular, Cech, and Borel-Moore), all classical cohomology groups (singular and Cech), and all finite-dimensional Hawaiian groups of X are trivial.