An oriented flat counterexample to the centre-rank formula for topological complexity

We construct a closed connected oriented flat five-manifold whose fundamental group has centre of rank two and whose unreduced topological complexity is at least ten. Since its universal cover is Euclidean space, this gives a negative answer to the proposed formula $\TC(M)=2\dim M+1-\rank Z(\pi_1M)$ for closed oriented aspherical manifolds. The lower bound follows from a nonzero product of nine degree-one zero divisors in the mod-$2$ cohomology ring of a real Bott manifold.