Erratum. (English) Zbl 1133.57303
In the author’s paper [“Constructing infinitely many smooth structures on \(3{\mathbb{CP}^2} \#n \overline{\mathbb{CP}}^{2}\)”, Math. Ann. 322, No. 2, 267–278 (2002; Zbl 0997.57042)] the proofs for the cases \(n=10\) and \(n=12\) are incomplete as they assume that the numerical Godeaux surface in [I. Dolgachev and C. Werner, J. Algeb. Geom. 8, 737–764 (1999; Zbl 0958.14023)] is simply connected as was claimed there. In [ibid. 10, 397 (2001; Zbl 0958.14023)] the authors pointed out that this proof does not hold.
In this erratum the author sketches an alternative proof for his theorem for the cases \(n=10\) and \(n=12\).
In this erratum the author sketches an alternative proof for his theorem for the cases \(n=10\) and \(n=12\).
MSC:
| 57R55 | Differentiable structures in differential topology |
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
| 57R57 | Applications of global analysis to structures on manifolds |
| 53D99 | Symplectic geometry, contact geometry |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
References:
| [1] | Dolgachev I. and Werner C. (1999). A simply connected numerical Godeaux surface with ample canonical class. J. Algebraic Geom. 8: 737-764 · Zbl 0958.14023 |
| [2] | Dolgachev I. and Werner C. (2001). Erratum to “A simply connected numerical Godeaux surface with ample canonical class”. J. Algebraic Geom. 10: 397 · Zbl 0958.14023 |
| [3] | Park B.D. (2000). Exotic smooth structures on \[3{\mathbb{CP}^2}\#n\overline{\mathbb{CP}}^2\] Proc. Am. Math. Soc. 128: 3057-3065 · Zbl 0957.57020 · doi:10.1090/S0002-9939-00-05357-0 |
| [4] | Park B.D. (2002). Constructing infinitely many smooth structures on \[3{\mathbb{CP}^2}\#n\overline{\mathbb{CP}}^2\] Math. Ann. 322: 267-278 · Zbl 0997.57042 · doi:10.1007/s002080100245 |
| [5] | Stipsicz A.I. and Szabó Z. (2006). Small exotic 4-manifolds with \[b_2^+=3\] Bull. Lond. Math. Soc. 38: 501-506 · Zbl 1094.57027 · doi:10.1112/S0024609306018406 |
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