The topology of the space of matrices of Barvinok rank two
2010 (English)In: Beiträge zur Algebra und Geometrie, ISSN 0138-4821, Vol. 51, no 2, 373-390 p.Article in journal (Refereed) Published
The Barvinok rank of a d x n matrix is the minimum number of points in Rd 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 x n matrices of Barvinok rank two. Let Bd,n denote this space modulo rescaling and translation. We show that Bd,n is a manifold, thereby settling a conjecture due to Develin. In fact, Bd,n is homeomorphic to the quotient of the product of spheres Sd-2 x Sn-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 Bd,n. Assuming d \ge n, for odd d the homology turns out to be isomorphic to that of Sd-2 x RPn-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 (Sd-2 x X)//Z2, where X is a finite cell complex of dimension at most d-2 admitting a free Z2-action.
Place, publisher, year, edition, pages
Lemgo, Germany: Heldermann Verlag , 2010. Vol. 51, no 2, 373-390 p.
Barvinok rank, tropical convexity, antipodal action, discrete Morse theory
Geometry Discrete Mathematics
IdentifiersURN: urn:nbn:se:kth:diva-70298ScopusID: 2-s2.0-77957035353OAI: oai:DiVA.org:kth-70298DiVA: diva2:486232
ProjectsSwedish Research Council Grant 2006-3279
QC 201202092012-01-302012-01-302012-02-09Bibliographically approved