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The homotopy type of the matroid Grassmannian. (English) Zbl 1076.52006

Ann. Math. (2) 158, No. 3, 929-952 (2003); erratum ibid. 170, No. 1, 493 (2009).
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 I. M. Gelfand and 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 [L. Anderson and 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. These stratifications have certain technical homotopy theoretic properties and are compatible with each other in a way that allows the induction to proceed.
As the author points out, this result “implies that the theory of matroid bundles is the same as the theory of vector bundles” and also suggests “that a CD manifold has the capacity to model many properties of smooth manifolds” beyond their use in the work of Gelfand and MacPherson. Forthcoming work of the author promises some developments in this direction. The paper is elegantly written and includes helpful motivational and background material.

MSC:

52C40 Oriented matroids in discrete geometry
55P10 Homotopy equivalences in algebraic topology
57R20 Characteristic classes and numbers in differential topology
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