The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs

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The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs evidence for this event

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The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs parent of this event

Budur, Fernandez de Bobadilla, Le, Nguyen, Cohomology of contact loci (arXiv:1911.08213v3) evidence for this event

Sampaio, On the Milnor fibres of initial forms of topologically equivalent holomorphic functions (arXiv:2503.17127) evidence for this event

With $f_1,f_2$ the Zariski counterexample germs, $\tilde f_j=f_j(z)+t^2$ on $\mathbb{C}^{N+1}$ are topologically right equivalent with isolated critical points, but $H_0((\tilde f_1)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}$ while $H_0((\tilde f_2)_*^{-1}(1);\mathbb{Z})\cong\mathbb{Z}^2$: one initial-form Milnor fibre is connected and the other has two components. This disproves both the embedded and the right-equivalence formulations. It says nothing about weaker invariants the conjecture was meant to capture, such as cohomology of contact loci. parent of this event