Maximal topological complexity of connected sums of aspherical manifolds

Let \(M\) and \(N\) be closed, connected, oriented aspherical manifolds of the same dimension \(n\geq 3\). We prove that their connected sum has ordinary unreduced topological complexity \(2n+1\). Thus joining two aspherical summands always attains the dimensional upper bound, although the connected sum is not required to be aspherical. The proof detects the two ordered cross products of the summands' fundamental classes in a quotient of the left-right join of their free-product fundamental group. A decomposition into factor and cyclic cosets shows that these mixed classes survive, and their survival obstructs a motion planner with \(2n\) domains.