An approach to the topological complexity of the Klein bottle

TL;DR

This paper discusses the topological complexity of the Klein bottle, comparing different approaches to proving its reduced topological complexity is 4, and clarifies the correct obstruction class.

Contribution

The paper presents a corrected approach to analyzing the topological complexity of the Klein bottle, emphasizing the simplicial structure and aligning with prior results.

Findings

The reduced topological complexity of the Klein bottle is 4.

The obstruction class associated with this complexity is nonzero.

Different proof methods can be reconciled after correction.

Abstract

Recently, Cohen and Vandembroucq proved that the reduced topological complexity of the Klein bottle is 4. Simultaneously and independently, we announced a proof of the same result. Mistakes were found in our argument, which was quite different than theirs. After correcting these, we found that our description of the obstruction class agreed with theirs. Our approach to showing that this obstruction is nonzero failed to do so, while theirs did not fail. Here we discuss our approach, which deals more directly with the simplicial structure of the Klein bottle.