A note on a topological approach to the $\mu$-constant problem in dimension 2

TL;DR

This paper demonstrates that analyzing homological and homotopical properties of cobordisms between graph manifolds does not suffice to prove the $$-constant conjecture in complex dimension 2, highlighting the limitations of current topological methods.

Contribution

It provides a counterexample showing the inadequacy of certain topological approaches to the $$-constant problem in complex dimension 2.

Findings

Homological and homotopical properties are insufficient for the conjecture.

Counterexample demonstrates limitations of current topological methods.

Studying cobordisms between graph manifolds does not resolve the $$-constant problem.

Abstract

We provide an example, which shows that studying homological and homotopical properties of cobordisms between arbitrary, that is not necessarily negative, graph manifolds is not enough to prove the $\mu$-constant conjecture of Le Dung Trang in complex dimension 2.