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.

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.

Reviewer: J.G.Oxley (Baton Rouge)

##### MSC:

05B35 | Combinatorial aspects of matroids and geometric lattices |

06C10 | Semimodular lattices, geometric lattices |

18G99 | Homological algebra in category theory, derived categories and functors |