Georgia Institute of Technology College of Sciences

Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope space of an oriented matroid based on Quillen filtration. We will prove the equality between their filtration (together with the induced maps), the topologically defined Kalinin filtration, and the combinatorially defined Varchenko-Gelfand dual degree filtration over Z/2Z. We will also explain how the dual degree filtration can serve as a Z-coefficient version of the other two in this setting. This is joint work with Kris Shaw.