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Homotopy \(K3\)’s with several symplectic structures. (English) Zbl 1067.57031
The main result states that for each integer \(n\), there is a homotopy \(K3\) smooth 4-manifold that admits \(n\) inequivalent symplectic structures. The construction of the manifold uses a generalization of the Fintushel-Stern link surgery, and the symplectic structures are distinguished by comparing their canonical classes.

57R57 Applications of global analysis to structures on manifolds
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R17 Symplectic and contact topology in high or arbitrary dimension
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
Full Text: DOI EMIS EuDML arXiv
[1] D Eisenbud, W Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies 110, Princeton University Press (1985) · Zbl 0628.57002
[2] R Fintushel, R J Stern, Knots, links, and 4-manifolds, Invent. Math. 134 (1998) 363 · Zbl 0914.57015
[3] R E Gompf, A I Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999) · Zbl 0933.57020
[4] C LeBrun, Diffeomorphisms, symplectic forms and Kodaira fibrations, Geom. Topol. 4 (2000) 451 · Zbl 0976.53091
[5] C T McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. \((4)\) 35 (2002) 153 · Zbl 1009.57021
[6] C T McMullen, C H Taubes, 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations, Math. Res. Lett. 6 (1999) 681 · Zbl 0964.53051
[7] W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467 · Zbl 0324.53031
[8] W P Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986) · Zbl 0585.57006
[9] S Vidussi, Smooth structure of some symplectic surfaces, Michigan Math. J. 49 (2001) 325 · Zbl 0992.57031
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