Grassmannians and Pseudosphere Arrangements

TL;DR

This paper introduces weighted pseudosphere arrangements as a topological extension of vector configurations, establishing homotopy equivalences with classical spaces in rank 3, and providing new classifying spaces for rank 3 vector bundles.

Contribution

It defines weighted pseudosphere arrangements and proves their homotopy equivalence to Grassmannians in rank 3, offering a new approach to classifying rank 3 vector bundles.

Findings

Homotopy equivalence between weighted pseudosphere arrangements and Grassmannians in rank 3

Contractibility of realization spaces for all rank 3 oriented matroids

Avoidance of real algebraic geometry difficulties in classifying rank 3 vector bundles

Abstract

We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements. These are defined as a weighted variant of arrangements of pseudospheres, as in the Topological Representation Theorem for oriented matroids. We show that in rank 3, the real Stiefel manifold, Grassmannian, and oriented Grassmannian are homotopy equivalent to the analogously defined spaces of weighted pseudosphere arrangements. As a consequence, this gives a new classifying space for rank 3 vector bundles and for rank 3 oriented vector bundles where the difficulties of real algebraic geometry that arrise in the Grassmannian can be avoided. In particular, we show for all rank 3 oriented matroids, that the subspace of weighted pseudosphere arrangements realizing that oriented matroid is contractible. This is a sharp contrast with vector configurations where the space of realizations can have the homotopy type of any primary real semialgebraic set.