Grassmannians and pseudosphere arrangements
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1225-1274.

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 arise 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 real semialgebraic set.

Nous étendons les configurations de vecteurs à des objets plus généraux, appelés arrangements de pseudo-sphères pondérées, aux propriétés combinatoires et topologiques plus agréables. Ils sont définis comme des variantes à poids d’arrangements de pseudo-sphères, tels qu’apparaissant dans le théorème de représentation topologique pour les matroïdes orientés. Nous montrons qu’en rang 3, la variété de Stiefel réelle, la grassmannienne et la grassmannienne orientée sont homotopes aux espaces définis de manière analogue pour les arrangements de pseudo-sphères pondérées. Par conséquent, cela définit de nouveaux espaces classifiants pour les fibrés vectoriels de rang 3 et les fibrés vectoriels orientés de rang 3 où les difficultés de géométrie algébriques soulevées par la grassmannienne peuvent être évitées. En particulier, nous montrons que pour tout matroïde orienté de rang 3, le sous-espace d’arrangements de pseudo-sphères pondérées qui le réalise est contractile. Cette situation contraste nettement avec celle des configurations de vecteurs, dont les espaces de réalisations peuvent avoir le type d’homotopie d’un ensemble semi-algébrique réel arbitraire.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.171
Classification: 52C30,  52C40,  14M15,  57R22
Keywords: Grassmannian, oriented matroid, pseudosphere arrangement, vector bundle
Michael Gene Dobbins 1

1 Department of Mathematical Sciences, Binghamton University (SUNY) Binghamton, New York, USA
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Michael Gene Dobbins. Grassmannians and pseudosphere arrangements. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1225-1274. doi : 10.5802/jep.171. https://jep.centre-mersenne.org/articles/10.5802/jep.171/

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