The homotopy type of the matroid Grassmannian
type: Item
instance of: scholarly article
title: The homotopy type of the matroid Grassmannian
zbMATH Open document ID: 1076.52006
author: Daniel K. Biss
published in: Annals of Mathematics. Second Series
publication date: 2004-05-27T00:00:00Z
review text: A {Combinatorial Differential} (CD) manifold is a simplicial complex together with certain auxiliary data. Examples arise whenever one has a smooth triangulation \(\eta: B \to M\) of a smooth \(k\)-manifold \(M\) (\(B\) is a simplicial complex), from which one may construct a map \(\widehat{B} \to \text{MacP}(k, n)\) from a refinement \(\widehat{B}\) of \(B\) to the so-called {MacPhersonian} \(\text{MacP}(k, n)\), a certain poset of oriented matroids. This map is an essentially combinatorial structure that -- as it turns out -- carries a great deal of information about the smooth manifold \(M\). Indeed, the origin of these constructions lies in work of Gelfand and MacPherson on combinatorial expressions for rational Pontrjagin characteristic classes [see \textit{I. M.~Gelfand} and \textit{R. D.~MacPherson}, Bull. Am. Math. Soc. 26, 304--309 (1992; Zbl 0756.57015)]. Generally, a CD manifold structure is an abstraction of this situation to one that does not {a priori} arise from a smooth triangulation of a manifold. In this CD setting, there is a theory of oriented matroid bundles that is analogous to the theory of vector bundles on smooth manifolds. In the former, an ``infinite MacPhersonian'' \(\text{MacP}(k, \infty)\) plays the role that the classifying space \(BO_k\) does in the latter. Let \(\text{G}(k, n)\) denote the (real) Grassmannian of \(k\)-planes in \(\mathbb{R}^n\) and let \(\| P\| \) denote the nerve of a poset \(P\). In [\textit{L.~Anderson} and \textit{J. F.~Davis}, Sel. Math. 8, 161--200 (2002; Zbl 1007.55010)], a certain map \(\pi : \text{G}(k, n) \to \| \text{MacP}(k, n)\| \) was shown to induce a split surjection on mod \(2\) cohomology. The main result of this paper is that \(\pi\) is actually a {homotopy equivalence}, for each \(1 \leq n \leq \infty\) and \(k \leq n\). The main idea of the proof is an induction, using a Schubert cell decomposition of the Grassmannian and an analogous stratification of the MacPhersonian that the author defines. T…
Mathematics Subject Classification ID: 52C40
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 2068686
zbMATH Keywords: oriented matroid
zbMATH Keywords: CD manifold
zbMATH Keywords: matroid bundle
zbMATH Keywords: Grassmannian
zbMATH Keywords: MacPhersonian
reviewed by: Gregory Lupton
MaRDI profile type: Publication
full work available at URL: annals.2003.158.929
OpenAlex ID: W2162456012
DOI: 10.4007/ANNALS.2003.158.929
description: scientific article; zbMATH DE number 2068686