Topological complexity, asphericity and connected sums

TL;DR

This paper establishes new lower bounds for the topological complexity of certain closed oriented manifolds based on cohomology and geometric properties, and fully determines the complexity for specific four-dimensional cases.

Contribution

It extends previous results by providing improved bounds on topological complexity for manifolds with particular cohomology classes and applies these to negatively curved and connected sum manifolds.

Findings

Topological complexity is at least 6 for odd dimensions under certain conditions.

In even dimensions, the bounds are at least 7 or 9, depending on the case.

In dimension four, the topological complexity of these connected sums is exactly 9.

Abstract

We show that if a closed oriented $n$-manifold $M$ has a non-trivial cohomology class of even degree $k$, whose all pullbacks to products of type $S^1\times N$ vanish, then the topological complexity $\mathrm{TC}(M)$ is at least $6$, if $n$ is odd, and at least $7$ or $9$, if $n$ is even. These bounds extend and improve a result of Mescher and apply for instance to negatively curved manifolds and to connected sums with at least one such summand. In fact, better bounds are obtained due to the non-vanishing of the Gromov norm. As a consequence, in dimension four, we completely determine the topological complexity of these connected sums, namely we show that it is equal to its maximum value nine. Furthermore, we discuss realisation of degree two homology classes by tori, and show how to construct non-realisable classes out of realisable classes in connected sums. The examples of this paper will quite often be aspherical manifolds whose fundamental groups have trivial center and connected sums. We thus discuss the possible relation between the maximum topological complexity $2n+1$ and the triviality of the center for aspherical $n$-manifolds and their connected sums.