Let \Delta be a Cohen-Macaulay complex

TL;DR

This paper discusses the properties and recent developments related to Cohen-Macaulay complexes, including Gorenstein* and homotopy Gorenstein* complexes, with connections to topology, combinatorics, and tropical geometry.

Contribution

It provides mathematical insights into Cohen-Macaulay complexes, explores their Gorenstein properties, and discusses recent combinatorial results motivated by tropical geometry.

Findings

Characterization of Gorenstein* and homotopy Gorenstein* complexes

Connections between Cohen-Macaulay complexes and the Poincaré conjecture

Results on homotopy Cohen-Macaulayness of geometric lattice subsets

Abstract

The concept of Cohen-Macaulay complexes emerged in the mid-1970s and swiftly became the focal point of an attractive and richly connected new area of mathematics, at the crossroads of combinatoics, commutative algebra and topology. As the main architect of these developments, Richard Stanley has made fundamental contributions over many years. This paper contains some brief mathematical discussions related to the Cohen-Macaulay property, and some personal memories. The characterization of Gorenstein* and homotopy Gorenstein* complexes and the relevance in that connection of the Poincar\'e conjecture is discussed. Another topic is combinatorial aspects of a recent result on the homotopy Cohen-Macaulayness of certain subsets of geometric lattices, motivated by questions in tropical geometry.