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Small exotic \(4\)-manifolds from lines and quadrics in \(\mathbb{CP}^{2}\)

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Abstract

We construct potentially new manifolds homeomorphic but not diffeomorphic to \(\mathbb{CP}^{2} \) # \(8\overline{{\mathbb{CP}}^{2}}\) and \(\mathbb{CP}^{2}\)# \(9\overline{{\mathbb{CP}}^{2}}\) via rational blowdown surgery along certain 4-valent plumbing graphs. This way all the graph classes from [5] have a representative which admits a rational blowdown leading to an exotic manifold. We emphasize the simplicity of the constructions which boils down to finding a good configuration of complex lines and quadrics in \(\mathbb{CP}^{2}\), and deciding which intersections to blow up.

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Acknowledgements

I would like to thank my advisor András Stipsicz for introducing me to smooth 4-dimensional topology, pointing me to the problems discussed in this paper, and selflessly guiding me through my PhD journey.

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  1. Department of Mathematics and its Applications at Central European University, Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13–15, Budapest, 1053, Hungary

    S. Mihajlović

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Mihajlović, S. Small exotic \(4\)-manifolds from lines and quadrics in \(\mathbb{CP}^{2}\). Acta Math. Hungar. 162, 584–603 (2020). https://doi.org/10.1007/s10474-020-01088-5

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  • DOI: https://doi.org/10.1007/s10474-020-01088-5

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