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The homology and shellability of matroids and geometric lattices. (English) Zbl 0772.05027
Matroid applications, Encycl. Math. Appl. 40, 226-283 (1992).
[For the entire collection see Zbl 0742.00052.]
This survey studies the properties of three simplicial complexes that can be associated with a matroid $$M$$: (i) the matroid complex of independent sets of $$M$$; (ii) the broken circuit complex relative to an ordering of the ground set $$E(M)$$; and (iii) the order complex that consists of chains of flats in the geometric lattice associated with $$M$$. The broken circuit complex (ii) consists of those subsets of $$E(M)$$ which do not contain a broken circuit, the latter being a set that is obtained from a circuit by removing its least element.
The notion of shellability is used by the author as a framework for his discussion. A complex is shellable if all its maximal faces are equicardinal and those maximal faces can be ordered in a certain way that is favorable for induction arguments. All three complexes listed above are shellable. The author aims to give a “unified and concise, yet gentle, introduction” to these three complexes and the links between them. The survey assumes a minimum of prerequisites and, for its first half, takes an entirely combinatorial approach. All algebraic aspects of the subject are left for the second half. There is a short section presenting the relevant parts of simplicial homology that is designed to make the paper basically self-contained. The survey concludes with some historical remarks and with numerous exercises that supplement the results covered in the body of the paper.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 06C10 Semimodular lattices, geometric lattices 18G99 Homological algebra in category theory, derived categories and functors
##### MathOverflow Questions:
Is this poset shellable?