Generalized infinite-order monodromy conjecture for surface singularity smoothings

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Let (V,p)(V,p) be a normal isolated surface singularity with rational homology 33-sphere link, and let it have a smoothing with Milnor fiber MM and monodromy ψ\psi. Suppose that ψ\psi acts with finite order on H∗(M,Z)H_\ast(M,\mathbb{Z}). Let pg(V,p)p_g(V,p) denote the geometric genus of the singularity. Generalized infinite-order monodromy conjecture. The monodromy ψ\psi has infinite order in the smooth mapping class group MCG⁡(M) \operatorname{MCG}(M)\, when pg(V,p)>0p_g(V,p)>0. This generalizes the hypersurface case and is motivated by examples for equivariant smoothings of quasi-homogeneous minimally elliptic surface singularities; the proposition in the paper gives finite order in the topological mapping class group under the stated homological hypothesis, while the smooth assertion remains open.

References

Primary source

Hokuto Konno, Jianfeng Lin, Anubhav Mukherjee and Juan Muñoz-Echániz, “On four-dimensional Dehn twists and Milnor fibrations”, arXiv:2409.11961 (2024).

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