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On the Donaldson polynomials of elliptic surfaces. (English) Zbl 0807.57016

The Donaldson invariants of smooth simply connected minimal elliptic surfaces over \(\mathbb{C} \mathbb{P}^ 1\) are polynomials in the intersection form \(q\) and canonical class \(k\). We compute, using Donaldson-Floer theory, the second nonzero coefficient in their expansion in \(q\) and \(k\) when the geometric genus is odd. We point out how this result (recently obtained also by Morgan and Mrowka, using different methods, for any geometric genus) yields the classification of smooth simply connected elliptic surfaces over \(\mathbb{C} \mathbb{P}^ 1\) up to diffeomorphism.
Reviewer: P.Lisca (Roma)

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
14J15 Moduli, classification: analytic theory; relations with modular forms
81T13 Yang-Mills and other gauge theories in quantum field theory
57R55 Differentiable structures in differential topology
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References:

[1] Atiyah, M.F.: New invariants for 3 and 4-dimensional manifolds. In: Wells, R.O. Jr. (ed.). The mathematical heritage of H. Weyl (Proc. Symp. Pure Math., Vol. 48, pp. 285-299) Providence, RI: Am. Math. Soc. 1988
[2] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc.77, 43-69 (1975) · Zbl 0297.58008
[3] Atiyah, M.F., Patodi, V. K., Singer, I.M.: Spectral asymmetry and R Riemannian geometry. II. Math. Proc. Camb. Philos. Soc.78, 405-432 (1975) · Zbl 0314.58016
[4] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Philos. Soc.79, 71-99 (1976) · Zbl 0325.58015
[5] Bauer, S.: Diffeomorphism types of elliptic surfaces withp g =1. Warwick preprint # 34 (1992)
[6] Braam, P.: Floer homology groups for homology 3-spheres. Adv. Math.88, 131-144 (1991) · Zbl 0856.57010
[7] Donaldson, S.K.: Polynomial invariants for smooth 4-manifolds. Topology29, 257-315 (1990) · Zbl 0715.57007
[8] Donaldson, S.K., Furuta, M.: On Floer homology. (in preparation)
[9] Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford: Clarendon 1990 · Zbl 0820.57002
[10] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys.118, 215-240 (1988) · Zbl 0684.53027
[11] Friedman, R., Morgan, J.W.: Smooth four-manifords and complex surfaces. (Ergeb. Math. Berlin Heidelberg New York: Springer (to appear)
[12] Fintushel, R., Stern, R.: Instanton homology of Seifert feibred homology three spheres. Proc. Lond. Math. Soc.61, 109-137 (1990) · Zbl 0705.57009
[13] Fintushel, R., Stern, R.: Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds. (Preprint) · Zbl 0843.57021
[14] Gompf, R.E.: Nuclei of elliptic surfaces. Topology30, 479-511 (1991) · Zbl 0732.57010
[15] Harer, J.: On handlebody structures for hypersurfaces in ?3 and ??3. Math. Ann.238, 51-58 (1978) · Zbl 0384.57016
[16] Kirby, R.: A calculus for framed links in S3. Invent. Math.45, 36-56 (1978) · Zbl 0377.55001
[17] Morgan, J.W., Mrowka, T.S.: On the diffeomorphic classification of regular elliptic surfaces. (Preprint) · Zbl 0807.57015
[18] Morgan, J.W., O’Grady, K.G.: Elliptic surfaces with p g =1:Smooth classification. (Lect. Notes Math., vol. 1545) Berlin Heidelberg New York: Springer 1992
[19] O’Grady, K.G.: Algebro-geometric analogues of Donaldson’s polynomials. Invent. Math.107, 351-395 (1992) · Zbl 0769.14008
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