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November 2015 $\mathrm{Pin}^{-} (2)$-monopole invariants
Nobuhiro Nakamura
J. Differential Geom. 101(3): 507-549 (November 2015). DOI: 10.4310/jdg/1445518922


We introduce a diffeomorphism invariant of 4-manifolds, the $\mathrm{Pin}^{-} (2)$-monopole invariant, defined by using the $\mathrm{Pin}^{-} (2)$-monopole equations. We compute the invariants of several 4-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connected sum of an elliptic surface $E(n)$ with arbitrary number of the 4-manifolds of the form of $S^2 \times \Sigma$ or $S^1 \times Y$ where $\Sigma$ is a compact Riemann surface with positive genus and $Y$ is a closed 3-manifold. As another application, we give an estimate of the genus of surfaces embedded in a 4-manifold $X$ representing a class $\alpha \in H_2 (X; l)$, where $l$ is a local coefficient on $X$.


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Nobuhiro Nakamura. "$\mathrm{Pin}^{-} (2)$-monopole invariants." J. Differential Geom. 101 (3) 507 - 549, November 2015.


Published: November 2015
First available in Project Euclid: 22 October 2015

zbMATH: 06537659
MathSciNet: MR3415770
Digital Object Identifier: 10.4310/jdg/1445518922

Rights: Copyright © 2015 Lehigh University


Vol.101 • No. 3 • November 2015
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