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Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends

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Abstract

We show 10/8-type inequalities for some end-periodic 4-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed 4-manifolds which do not admit positive scalar curvature metrics.

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Acknowledgements

The authors would like to express their deep gratitude to Yukio Kametani for answering their many questions on his preprint [12]. The authors would also like to express their appreciation to Mikio Furuta for informing them of Kametani’s preprint and encouragements on this work. The authors would like to express their deep gratitude to Danny Ruberman for giving comments on examples of this paper. The authors also wish to thank Andrei Teleman for informing them of Veloso’s argument [38] and answering their questions on it. The authors would also like to express their appreciation to Jianfeng Lin and Fuquan Fang for pointing out the relation between our work and that of Schoen and Yau [34]. The authors also appreciate Ko Ohashi’s, Mayuko Yamashita’s, Kyungbae Park’s and Kouki Sato’s helpful comments on the paper [9], on equivariant KO-theory, homology cobordisms and examples of homology \(S^1\times S^3\)’s respectively. The first author was supported by JSPS KAKENHI Grant Numbers 16J05569 and 19K23412. The second author was supported by RIKEN iTHEMS Program and JSPS KAKENHI Grant Number 17J04364. Both authors were supported by JSPS Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area) No.17H06461 and the Program for Leading Graduate Schools, MEXT, Japan.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Hokuto Konno

  2. RIKEN, the Institute of Physical and Chemical Research, 2-1 Hirosawa Wako, Saitama, 351-0198, Japan

    Masaki Taniguchi

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  1. Hokuto Konno
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  2. Masaki Taniguchi
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Correspondence to Masaki Taniguchi.

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Konno, H., Taniguchi, M. Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends. Invent. math. 222, 833–880 (2020). https://doi.org/10.1007/s00222-020-00979-2

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  • DOI: https://doi.org/10.1007/s00222-020-00979-2

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