On the Sternfeld-Levin counterexamples to a conjecture of Chogoshvili-Pontrjagin

An inconclusive proof in a 1937 paper by G. Chogoshvili spawned an interesting dimensiontheoretic conjecture which we call the Chogoshvili-Pontrjagin Conjecture. In 1991, Y. Sternfeld found an ingenious counterexample to this conjecture which he and M. Levin greatly generalized in 1995. In this note we point out a previously unobserved property of the Sternfeld-Levin examples, and we reinterpret their significance in light of this property. Also, we present a version of the Levin-Sternfeld proof which is more “topological” and less “lattice-theoretic” than the original.