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The elliptic genus on non-spin even 4-manifolds. (English) Zbl 1112.58025

The rigidity under circle actions of the elliptic genus on oriented non-spin closed smooth 4-manifolds with even intersection form is proved. See also S. Ochanine, Topology 26, 143–151 (1987; Zbl 0626.57014), C. H. Taubes, Commun. Math. Phys. 122, No. 3, 455–526 (1989; Zbl 0683.58043), D. Acosta and T. Lawson, Enseign. Math. 43, No. 1–2, 27–32 (1997; Zbl 0889.57013).

MSC:

58J26 Elliptic genera
57R91 Equivariant algebraic topology of manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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References:

[1] Acosta, D.; Lawson, T., Even non-spin manifolds, \(spin^c\) structures, and duality, Enseign. Math. (2), 43, 1-2, 27-32 (1997) · Zbl 0889.57013
[2] Atiyah, M. F.; Bott, R., The Lefschetz fixed point theorem for elliptic complexes. II, Appl. Ann. Math., 88, 451-491 (1968) · Zbl 0167.21703
[3] Atiyah, M. F.; Hirzebruch, F., Spin manifolds and group actions, (Essays in Topology and Related Subjects (1970), Springer: Springer Berlin), 18-28 · Zbl 0193.52401
[4] Atiyah, M. F.; Segal, G., The index of elliptic operators II, Ann. of Math., 86, 531-545 (1968) · Zbl 0164.24201
[5] Bott, R.; Taubes, T., On the rigidity theorems of Witten, J. AMS, 2, 1, 137-186 (1989) · Zbl 0667.57009
[6] Donaldson, S. K., An application of gauge theory to four-dimensional topology, J. Differential Geom., 18, 2, 279-315 (1983) · Zbl 0507.57010
[7] Herrera, H.; Herrera, R., Generalized elliptic genus and cobordism class of nonspin real Grassmannians, Ann. Global Anal. Geom., 24, 4, 323-335 (2003) · Zbl 1045.57016
[8] Hirzebruch, F.; Berger, T.; Jung, R., Manifolds and Modular Forms. Aspects of Mathematics (1992), Vieweg
[9] Hirzebruch, F.; Slodowy, P., Elliptic genera, involutions, and homogeneous spin manifolds, Geometriae Dedicata, 35, 309-343 (1990) · Zbl 0712.57010
[10] Huck, W.; Puppe, V., Circle actions on 4-manifolds. II, Arch. Math. (Basel), 71, 6, 493-500 (1998) · Zbl 0926.57010
[11] Ochanine, S., Sur les genres multiplicatifs définis par des intégrales elliptiques, Topology, 26, 143-151 (1987) · Zbl 0626.57014
[12] Taubes, C. H., \(S^1\) actions and elliptic genera, Comm. Math. Phys., 122, 3, 455-526 (1989) · Zbl 0683.58043
[13] Witten, E., The index of the Dirac operator on loop space, (Landweber, P. S., Elliptic Curves and Modular Forms in Algebraic Topology. Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes Math., vol. 1326 (1988), Springer: Springer Berlin), 161-181
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