Filtrations of Tope Spaces of Oriented Matroids

We compare three filtrations of the tope space of an oriented matroid. The first is the dual Varchenko-Gelfand degree filtration, the second filtration is from Kalinin’s spectral sequence, and the last one derives from Quillen’s augmentation filtration. We show that all three filtrations and the respective maps coincide over $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. We also show that the dual Varchenko-Gelfand degree filtration can be made into a filtration of the $\mathbb{Z}$-sign cosheaf on the fan of the underlying matroid. This was previously carried out with $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-coefficients by the first author and Renaudineau using the Quillen filtration and has applications to real algebraic geometry via patchworking.