The topology of the space of matrices of Barvinok rank two

The Barvinok rank of a $d \times n$ matrix is the minimum number of points in $\mathbb{R}^d$ such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real $d \times n$ matrices of Barvinok rank two. Let $B_{d,n}$ denote this space modulo rescaling and translation. We show that $B_{d,n}$ is a manifold, thereby settling a conjecture due to Develin. In fact, $B_{d,n}$ is homeomorphic to the quotient of the product of spheres $S^{d-2} \times S^{n-2}$ under the involution which sends each point to its antipode simultaneously in both components. In addition, using discrete Morse theory, we compute the integral homology of $B_{d,n}$. Assuming $d \ge n$, for odd $d$ the homology turns out to be isomorphic to that of $S^{d-2} \times \mathbb{RP}^{n-2}$. This is true also for even $d$ up to degree $d-3$, but the two cases differ from degree $d-2$ and up. The homology computation straightforwardly extends to more general complexes of the form $(S^{d-2} \times X)/\mathbb{Z}_2$, where $X$ is a finite cell complex of dimension at most $d-2$ admitting a free $\mathbb{Z}_2$-action.