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Le–Ramanujam product cobordism problem in complex dimension two

Determine whether, in general, for n = 2 and a μ-constant deformation f_s of a complex polynomial with an isolated critical point at the origin, the natural integral homology cobordism W = M0 \ Ms between the Milnor fibers M0 and Ms is homeomorphic to a product cobordism.

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Background

For a μ-constant deformation of isolated hypersurface singularities, Le–Ramanujam showed that when n ≠ 2 the cobordism between Milnor fibers is diffeomorphic to a product. In complex dimension n = 2, the corresponding cobordism W is an integral homology cobordism.

It remains unresolved whether W is homeomorphic to a product in dimension two, a problem commonly referred to as the Le–Ramanujam problem.

References

When $n = 2$ the cobordism $W = M_0 \setminus M_s$ is an integral homology cobordism and it is an open problem whether, in general, $W$ is homeomorphic to a product cobordism (this is the famous Le--Ramanujam problem).

— The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory (2411.12202 - Konno et al., 19 Nov 2024) in Remark, Section 2.3 (μ-constant deformations)