2134/34775 Alexey Bolsinov Alexey Bolsinov Anton Izosimov Anton Izosimov Smooth invariants of focus-focus singularities and obstructions to product decomposition Loughborough University 2018 untagged Mathematical Sciences not elsewhere classified 2018-09-10 10:02:30 Journal contribution https://repository.lboro.ac.uk/articles/journal_contribution/Smooth_invariants_of_focus-focus_singularities_and_obstructions_to_product_decomposition/9376784 We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic 4-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus-focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung’s conjecture which says that any non-degenerate singularity can be smoothly decomposed into an almost direct product of standard singularities.