Topological Complexity of the Klein bottle

TL;DR

This paper determines that the topological complexity of the Klein bottle and non-orientable surfaces of genus at least 2 is exactly 4, completing recent research in this area.

Contribution

It establishes the exact topological complexity for the Klein bottle and all non-orientable surfaces with genus ≥ 2, extending previous work by Dranishnikov.

Findings

Topological complexity of the Klein bottle is 4.

Non-orientable surfaces of genus ≥ 2 also have topological complexity 4.

Completes the classification of topological complexity for these surfaces.

Abstract

We show that the (normalized) topological complexity of the Klein bottle is $4$. We also show that, for any $g\geq 2$, $TC(N_g)=4$. This completes the recent work by Dranishnikov on the topological complexity of non-orientable surfaces.