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\(\text{ Pin }^-(2)\)-monopole equations and intersection forms with local coefficients of four-manifolds

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Abstract

We introduce a variant of the Seiberg-Witten equations, \(\text{ Pin }^-(2)\)-monopole equations, and give its applications to intersection forms with local coefficients of four-manifolds. The first application is an analogue of Froyshov’s results on four-manifolds with definite intersection forms with local coefficients. The second is a local coefficient version of Furuta’s \(10/8\)-inequality. As a corollary, we construct nonsmoothable spin four-manifolds satisfying Rohlin’s theorem and the \(10/8\)-inequality.

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Acknowledgments

The author would like to express his deep gratitude to the referee for his detailed and valuable comments including a long list of suggestions over 40 items which enable the author to improve the paper drastically. It is also a pleasure to thank M. Furuta, Y. Kametani, K. Kiyono and S. Matsuo for helpful discussions and their comments on the earlier versions of the paper.

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  1. Department of Mathematics, Gakushuin University, 1-5-1, Mejiro, Toshima-ku, Tokyo, 171- 8588, Japan

    Nobuhiro Nakamura

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  1. Nobuhiro Nakamura
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Correspondence to Nobuhiro Nakamura.

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Nakamura, N. \(\text{ Pin }^-(2)\)-monopole equations and intersection forms with local coefficients of four-manifolds. Math. Ann. 357, 915–939 (2013). https://doi.org/10.1007/s00208-013-0924-3

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