Grassmannians and pseudosphere arrangements

Dobbins, Michael Gene

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1225-1274.

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.

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.

Classification : 52C30, 52C40, 14M15, 57R22
Keywords: Grassmannian, oriented matroid, pseudosphere arrangement, vector bundle
Mot clés : Grassmannienne, matroïde orienté, arrangement de pseudosphères, fibré vectoriel

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