Links of complex analytic singularities

TL;DR

This paper reviews and enhances the understanding of the local topology of complex analytic spaces, focusing on links of singularities, and introduces a new characterization for Cohen-Macaulay cases.

Contribution

It strengthens previous results on fundamental groups of links of isolated singularities and characterizes these groups for Cohen-Macaulay singularities.

Findings

Strengthened results on fundamental groups of links

Characterization of links of Cohen-Macaulay singularities

Compilation of open problems and conjectures

Abstract

This is a part survey part research paper studying the local topology of complex analytic spaces. We review and strengthen the results of Kapovich--Koll\'ar "Fundamental groups of links of isolated singularities" (1109.4047) and incorporate the paper "Dual graphs of exceptional divisors" (1203.2640) into the current one. A new result is the characterization of the fundamental group of links of Cohen-Macaulay singularities. The last section lists numerous open problems and conjectures. Version 2: Questions 62--65 revised following comments of Shaneson and Maxim. Version 3: Many small changes, especially in section about Questions and Problems.