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Topological triviality of mu-constant families of surface singularities

OpenAI · September 24, 2026

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Abstract

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

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Possibly relevant literature (found by the agent, not checked)

Maciej Borodzik and Stefan Friedl, A note on a topological approach to the μ-constant problem in dimension 2 (arXiv:1304.1363, April 2013; Revista Matemática Complutense 27, 345–350, 2014). Shows that certain homological and homotopical properties of cobordisms between arbitrary graph manifolds do not suffice to settle the dimension-2 case, while emphasizing the additional negative-definiteness condition satisfied by singularity links; this bears on the introduction's statement that the classical cobordism argument does not apply to surfaces.
Maximiliano Leyton-Álvarez, Hussein Mourtada and Mark Spivakovsky, Newton non-degenerate μ-constant deformations admit simultaneous embedded resolutions (arXiv:2001.10316; Compositio Math. 158 (2022), 1268–1297). Shows that a Newton non-degenerate deformation is μ-constant if and only if it admits a simultaneous embedded resolution, which gives topological triviality; this is a special case of Theorem 1.1, also proved through simultaneous resolution.
R. Giménez Conejero, Andreas Lind and Aurélio Menegon, Embedded topological triviality of separable families of singularities (arXiv:2511.23185, November 2025). Gives sufficient conditions for embedded topological triviality of families of function germs, with applications to a broad class of μ-constant deformations; this overlaps in part with the conclusion of Theorem 1.1.
Aurelio Menegon Neto, On the topology of families of isolated singularities (arXiv:1411.7609, November 2014; withdrawn in v2). Claimed a solution of the μ-constant problem in all dimensions via uniform Milnor radii; the author withdrew it "due to a crucial error". This bears on the uniform-radius approach that the paper contrasts with its own when discussing Peters (2026).

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 064

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Cite

@misc{OAI:Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026,
  author = {{OpenAI}},
  title = {{Topological triviality of $\mu$-constant families of surface singularities}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf}{OAI:Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026}},
  year = {2026}
}