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. $\pi_n(X)$ 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.

💡 Research Summary

The paper presents a striking counterexample in geometric topology: a Peano continuum (X) that is a one‑point compactification of a polyhedron, is weakly homotopy equivalent to a point (all homotopy groups (\pi_n(X)) vanish), yet is not contractible. Moreover, (X) enjoys both homological and cohomological local connectedness (it is an HLC and clc space). The authors also show that every classical homology theory (singular, Čech, Borel‑Moore) and every classical cohomology theory (singular, Čech) yields trivial groups for (X), and that all finite‑dimensional Hawaiian groups of (X) are zero.

The construction proceeds by taking an infinite sequence of polyhedral pieces whose diameters tend to zero, arranging them in a telescope‑like fashion, and then adding a single “point at infinity’’ to obtain a compact space. Each bonding map in the telescope is a cell‑like map; such maps preserve shape while killing homotopy groups in the inverse limit. Consequently, the inverse limit space (X) has (\pi_n(X)=0) for every (n\ge0). The authors verify this by explicit analysis of the induced maps on homotopy groups and by invoking shape‑theoretic arguments that guarantee weak homotopy equivalence to a point.

To demonstrate non‑contractibility, two independent arguments are given. First, removing the point at infinity leaves a space that fails to be locally connected, a property incompatible with contractibility in compact metric spaces. Second, the shape of (X) is shown to be non‑trivial: although each finite stage has the shape of a point, the inverse limit retains a “wild’’ shape reminiscent of the Hawaiian earring. Using the Steenrod–Švarc–Whitehead theorem, the authors prove that no homotopy contracting (X) to a point can exist, because any such contraction would induce a shape equivalence that contradicts the observed non‑trivial shape.

The paper also establishes that (X) is HLC and clc. This follows from the fact that each polyhedral stage is locally homologically and cohomologically connected, and the inverse limit of HLC (resp. clc) spaces under cell‑like bonding maps preserves these local properties. As a result, all singular, Čech, and Borel‑Moore homology groups of (X) vanish, a fact proved via Mayer–Vietoris sequences and direct chain‑complex calculations that exploit the shrinking nature of the telescope.

Finally, the authors address the finite‑dimensional Hawaiian groups (\mathcal{H}_n(X)), which capture higher‑order “wild’’ homotopy phenomena not detected by ordinary (\pi_n). By showing that every loop or sphere representing a potential non‑trivial element can be homotoped into an arbitrarily small stage of the telescope, they prove that each Hawaiian group collapses to the trivial group. This argument relies on delicate control of the bonding maps and the smallness condition inherent in the construction.

In summary, the work demonstrates that vanishing of all homotopy groups, all classical homology and cohomology groups, and even all finite‑dimensional Hawaiian groups does not guarantee contractibility when the space is not a CW‑complex. Moreover, it shows that a space can be locally well‑behaved (HLC, clc) while still exhibiting global non‑contractibility. The example enriches our understanding of the limits of classical algebraic invariants and highlights the subtle interplay between shape theory, cell‑like maps, and wild topological constructions.