Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends
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Positive scalar curvature and 10/8-type inequalities on 4-manifolds with periodic ends Hokuto Konno1 · Masaki Taniguchi2
Received: 24 March 2019 / Accepted: 7 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . 2.2 Mrowka–Ruberman–Saveliev invariant and Lin’s formula 2.3 Kametani’s theorem . . . . . . . . . . . . . . . . . . . . 2.4 Moduli theory . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . 3.1 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kuranishi model . . . . . . . . . . . . . . . . . . . . . . 3.3 Spin -structure on the Seiberg–Witten moduli space . . 3.4 Main construction . . . . . . . . . . . . . . . . . . . . .
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B Masaki Taniguchi
[email protected] Hokuto Konno [email protected]
1
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
2
RIKEN, the Institute of Physical and Chemical Research, 2-1 Hirosawa Wako, Saitama 351-0198, Japan
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H. Konno, M. Taniguchi 3.5 Completion of the proof of Theorem 1.1 . . . . . . 4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Examples of homology S 1 × S 3 ’s . . . . . . . . . 4.1.1 Examples from algebraically canceling pairs 4.1.2 Examples from 2-knots . . . . . . . . . . . . 4.2 Examples of homology 3-spheres with ψ > 0 . . . 4.3 Homology S 1 × S 3 ’s admitting no PSC metrics . . 4.3.1 Results for X (Y, H ) . . . . . . . . . . . . . 4.3.2 Connected sum along S 1 . . . . . . . . . . . 4.3.3 Results for surgeries of 2-knots . . . . . . . . 4.4 Comparison with other methods . . . . . . . . . . 4.4.1 Dirac obstruction . . . . . . . . . . . . . . . 4.4.2 Enlargeability . . . . . . . . . . . . . . . . . 4.4.3 Schoen–Yau’s method . . . . . . . . . . . . 4.4.4 Lin’s formula . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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