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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 allcolumns of the matrix. The concept originated in work by Barvinok and others onthe 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 modulorescaling and translation. We show that $B_{d,n}$ is a manifold, therebysettling a conjecture due to Develin. In fact, $B_{d,n}$ is homeomorphic to thequotient of the product of spheres $S^{d-2} \times S^{n-2}$ under theinvolution which sends each point to its antipode simultaneously in bothcomponents. In addition, using discrete Morse theory, we compute the integralhomology of $B_{d,n}$. Assuming $d \ge n$, for odd $d$ the homology turns outto be isomorphic to that of $S^{d-2} \times \mathbb{RP}^{n-2}$. This is truealso 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 moregeneral complexes of the form $(S^{d-2} \times X)/\mathbb{Z}_2$, where $X$ is afinite cell complex of dimension at most $d-2$ admitting a free$\mathbb{Z}_2$-action.

The topology of the space of matrices of Barvinok rank two