On irreducible simply-connected 4-manifolds. (English) Zbl 0881.57016
In the last few years, based on results of Donaldson, Gompf, Mrowka and Taubes, see [S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds (1990; Zbl 0820.57002); R. E. Gompf, Ann. Math., II. Ser. 142, No. 3, 527-595 (1995; Zbl 0849.53027); R. E. Gompf and T. S. Mrowka, ibid. 138, No. 1, 61-111 (1993; Zbl 0805.57012); C. H. Taubes, Math. Res. Lett. 1, No. 6, 809-822 (1994; Zbl 0853.57019); ibid. 2, No. 1, 9-13 (1995; Zbl 0854.57019)], an ambitious classification scheme emerged regarding smooth simply-connected closed 4-manifolds:
Minimal Conjecture [C. H. Taubes, in the I. International Press Lectures, UC Irvine, March 1996; D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes), Sémin. Bourbaki, Vol. 1995/96, Exp. No. 812, 195-220 (1997)]. Every simply-connected smooth closed 4-manifold \(X\) can be decomposed as \(X=X_1\# \cdots \# X_n\), where \(X_i\) are symplectic 4-manifolds with both the symplectic and the opposite orientations allowed, and where \(S^4\) corresponds to the empty sum.
The first counterexamples to the Minimal Conjecture were constructed by the author in [Simply connected irreducible 4-manifolds with no symplectic structures (Preprint 1996)], and were later generalized in [Z. Szabo, Math. Res. Lett. 3, No. 6, 731-741 (1996; Zbl 0874.57021)] and by R. Fintushel and R. Stern in [Knots, links and 4-manifolds (Preprint 1996)].
This is an expository paper, that aims to give a short account of the author’s counterexamples [loc. cit.], for a detailed description see there. The paper also contains additional examples, and gives an interesting family of irreducible simply-connected smooth closed 4-manifolds with small signature, presented in Section 3. (A simply-connected smooth closed 4-manifold is irreducible if any connected sum decomposition satisfies that one of the summands is a homotopy \(S^4)\).
Minimal Conjecture [C. H. Taubes, in the I. International Press Lectures, UC Irvine, March 1996; D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes), Sémin. Bourbaki, Vol. 1995/96, Exp. No. 812, 195-220 (1997)]. Every simply-connected smooth closed 4-manifold \(X\) can be decomposed as \(X=X_1\# \cdots \# X_n\), where \(X_i\) are symplectic 4-manifolds with both the symplectic and the opposite orientations allowed, and where \(S^4\) corresponds to the empty sum.
The first counterexamples to the Minimal Conjecture were constructed by the author in [Simply connected irreducible 4-manifolds with no symplectic structures (Preprint 1996)], and were later generalized in [Z. Szabo, Math. Res. Lett. 3, No. 6, 731-741 (1996; Zbl 0874.57021)] and by R. Fintushel and R. Stern in [Knots, links and 4-manifolds (Preprint 1996)].
This is an expository paper, that aims to give a short account of the author’s counterexamples [loc. cit.], for a detailed description see there. The paper also contains additional examples, and gives an interesting family of irreducible simply-connected smooth closed 4-manifolds with small signature, presented in Section 3. (A simply-connected smooth closed 4-manifold is irreducible if any connected sum decomposition satisfies that one of the summands is a homotopy \(S^4)\).
MSC:
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
