## On manifolds homeomorphic to $${\mathbb{C}}P^ 2\#8\overline{{\mathbb{C}}P}^ 2$$.(English)Zbl 0691.57008

The following problem has been of considerable interest recently: Which is the smallest value of k (denote it by $$k_ 0)$$ such that there is a complex surface homeomorphic but not diffeomorphic to $${\mathbb{C}}P^ 2\#k\overline{{\mathbb{C}}P}^ 2$$, equipped with the standard smooth structure? It was shown earlier by R. Friedman and J. W. Morgan [J. Differ. Geom. 27, 297-369 (1988; Zbl 0669.57016)] and, independently, by C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)] that $$k_ 0\leq 9$$. The main result of the paper under review is that $$k_ 0\leq 8$$. Namely, it is demonstrated that the Barlow surface is not diffeomorphic to $${\mathbb{C}}P^ 2\#8\overline{{\mathbb{C}}P}^ 2$$ (although these two 4-manifolds are homeomorphic by M. H. Freedman’s classification theorem). A key ingredient of the proof is a new invariant $$\phi$$ defined for oriented Riemannian 4-manifolds X such that $$b_ 1(X)=0$$, $$b^+_ 2(X)=1$$, and $$b^-_ 2(X)=8k$$, which is simpler than S. K. Donaldson’s $$\Gamma_ X$$-invariant and polynomial invariants.
Reviewer: D.Repovš

### MSC:

 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

### Citations:

Zbl 0669.57016; Zbl 0613.14018
Full Text:

### References:

 [1] [AHS] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. London, Ser. A362, 425-461 (1978) · Zbl 0389.53011 [2] [B] Barlow, R.N.: A simply connected surface of general type withp g=0. Invent. Math.79, 293-301 (1985) · Zbl 0561.14015 [3] [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0718.14023 [4] [D1] Donaldson, S.K.: Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc.50, 1-26 (1985) · Zbl 0547.53019 [5] [D2] Donaldson, S.K.: Irrationality and theh-cobordism conjecture. J. Differ. Geom.26, 141-168 (1987) · Zbl 0631.57010 [6] [D3] Donaldson, S.K.: The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differ. Geom.26, 397-428 (1987) · Zbl 0683.57005 [7] [D4] Donaldson, S.K.: Polynomial invariants for smooth four-manifolds. Topology (to appear) [8] [FS] Fintushel, R., Stern, R.J.:SO(3)-connections and the topology of 4-manifolds. J. Differ. Geom.20, 523-539 (1984) · Zbl 0562.53023 [9] [FU] Freed, D.S., Uhlenbeck, K.K.: Instantons and Four-manifolds. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0559.57001 [10] [FM] Friedman, R., Morgan, J.W.: On the diffeomorphism types of certain algebraic surfaces I. J. Differ. Geom.27, 297-369 (1988) · Zbl 0669.57016 [11] [GH] Griffiths, P., Harris, J.: Residues and zero-cycles on algebraic varieties. Ann. Math.108, 461-505 (1978) · Zbl 0423.14001 [12] [GZ] van der Geer, G. Zagier, D.: The Hilbert Modular group for the Field $$Q\left( {\sqrt {13} } \right)$$ . Invent Math.42, 93-133 (1977) · Zbl 0366.10024 [13] [HK] Hambleton, I., Kreck, M.: On the Classification of Topological 4-Manifolds with Finite Fundamental Group. Math. Ann.280, 1-20 (1988) · Zbl 0616.57009 [14] [K] Kotschick, D.: Oxford Thesis 1989 [15] [OV] Okonek, C., Van de Ven, A.: Stable bundles and differentiable structures on certain algebraic surfaces. Invent. Math.86, 357-370 (1986) · Zbl 0613.14018 [16] [OV2] Okonek, C., Van de Ven, A.:T-type-invariants associated toPU(2)-bundles and the differentiable structure of Barlow’s surface. Invent. Math.95, 601-614 (1989) · Zbl 0691.57007 [17] [R] Ried, M.: Surfaces withp g=0.K 2=1. J. Fac. Sci. Univ. Tokyo Sec. IA,25, (No. 1) 75-92 (1978) [18] [S] Schwarzenberger, R.L.E.: Vector bundles on algebraic surfaces. Proc. London Math. Soc. (3)11, 601-622 (1961) · Zbl 0212.26003 [19] [T] Tyurin, A.N.: Cycles, curves and vector bundles on an algebraic surface. Duke Math. J.54, 1-26 (1987) · Zbl 0631.14009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.