##
**Irrationality and the h-cobordism conjecture.**
*(English)*
Zbl 0631.57010

This paper contains the details of the author’s earlier results announced in “La topologie différentielle des surfaces complexes [C. R. Acad. Sci., Paris, Sér. I 301, 317-320 (1985; Zbl 0584.57010)]. The main theorem is the construction of a very surprising and powerful invariant. This is carried out in section II where the author constructs a diffeomorphism invariant \(\Gamma_ X\) for smooth, closed oriented 4- manifolds X whose intersection form has type (1,n) \((b^+_ 2(X)=1)\). Let \(\Omega =\{\theta \in H^ 2(X;{\mathbb{R}})| \theta \cdot \theta >0\}\) be the positive cone of the intersection form. For \(e\in H^ 2(X;{\mathbb{Z}})\) with \(e\cdot e=-1\) let \(w_ e=e^{\perp}\cap \Omega\), where \(e^{\perp}\subset H^ 2(X;{\mathbb{R}})\). \({\mathcal C}_ x=\) set of connected components of \(\Omega\)-(\(\cup_{e}W_ e)\). Then \(\Gamma_ X\) is a map \(\Gamma_ x: {\mathcal C}_ x\to H^ 2(X;{\mathbb{Z}})\) such that (i) \(\Gamma_ x(-C)=-\Gamma_ x(C)\); (ii) if \(C_{-1}\) and \(C_ 0\) are chambers in the same component of \(\Omega\) and if a path between \(C_{- 1}\) and \(C_ 0\) meets walls \(W_{e_ 1},...,W_{e_ k}\), where \(e_ i\cdot C_{-1}<0<e_ i\cdot C_ 1\), then \(\Gamma_ X(C_ 1)=\Gamma_ X(C_{-1})+2\sum^{k}_{i=1}e_ i\); (iii) if \(f: X_ 1\to X_ 2\) is an orientation preserving diffeomorphism then \(\Gamma_ x(f^*(C))=f^*(\Gamma_ x(C)).\)

Given the methods and results about the moduli space of anti-self-dual (ASD) connections obtained by Donaldson, Taubes and Freed-Uhlenbeck the author constructs, for a generic metric g on X and a harmonic form \(\omega\) representing an element of \({\mathcal C}_ x\), a class \(\Gamma\) (g,\(\omega)\) in \(H^ 2(X;{\mathbb{Z}})\). More precisely he fixes an SU(2)- bundle with \(c_ 2=1\) and considers the moduli space of (ASD)- connections M(g)\(\subset B\), the space of gauge equivalence of connections. Then for (g,\(\omega)\) as above the author had constructed in a previous paper a universal bundle E over \(B^*\times X\), \(B^*\) the irreducible connections. By the slant product with \(c_ 2(E)\) one obtains, after passing to the adjoint map, a map \(\mu\) : \(H^ 2(X;{\mathbb{Z}})\to H^ 2(B^*;{\mathbb{Z}})\). If M(g) were compact one could take the Kronecker product with the fundamental homology class [M(g)] in \(H_ 2(B^*;{\mathbb{Z}})\) from the author’s paper [J. Differ. Geom. 26, 397-428 (1987)] to obtain \(\Gamma\) (g,w). In general he identifies a correction term associated with the description of the end of M(g), which will give a class in \(H^ 2(X;{\mathbb{Z}})\) when M(g) is not compact. This is the content of Proposition 2.9. The proof of the main theorem mentioned above (2.15) is finished by studying the variation of g and \(\omega\).

In general it is at the moment very difficult to compute the Donaldson invariant but if X happens to be an algebraic surface it is closely related to stable vector bundles over it. This is carried out in part (a) of section III. In part (b) he applies this to the Dolgachev surfaces \(D_{p,q}\) which are obtained from the elliptic surface Y diffeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) by logarithmic transformation of multiplicity p and q in two nonsingular fibres. If \((p,q)=1\) these surfaces are homeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) by Freedman’s classification of 1-connected 4-manifolds [M. Freedman, ibid. 17, 357-453 (1982; Zbl 0528.57011)]. But the author uses his invariant combined with information about stable vector bundles to show that \(D_{2,3}\) is not diffeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) (Theorem 3.24). (These computations were later on generalized by R. Friedman and J. Morgan [ibid. 27, 297-369 (1988)], C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)], and others.)

In section IV (Appendix) the author indicates how his results extend to take account of fundamental groups. As an application he proves (Cor. 4.2) that there is no self-diffeomorphism f on the Godeaux surface S (the orbit space of the quintic in \({\mathbb{C}}P^ 3\) by the obvious \({\mathbb{Z}}_ 5\)-action) such that \(f^*K=K\otimes L^ i\), \(i=1,2,3,4\). Here K is the canonical line bundle and L the flat complex line bundle associated with the covering. (In an article by I. Hambleton and the reviewer [Math. Ann. 280, 85-104 (1988; Zbl 0616.57009)] it was later on shown that these maps do exist as homeomorphisms.)

Given the methods and results about the moduli space of anti-self-dual (ASD) connections obtained by Donaldson, Taubes and Freed-Uhlenbeck the author constructs, for a generic metric g on X and a harmonic form \(\omega\) representing an element of \({\mathcal C}_ x\), a class \(\Gamma\) (g,\(\omega)\) in \(H^ 2(X;{\mathbb{Z}})\). More precisely he fixes an SU(2)- bundle with \(c_ 2=1\) and considers the moduli space of (ASD)- connections M(g)\(\subset B\), the space of gauge equivalence of connections. Then for (g,\(\omega)\) as above the author had constructed in a previous paper a universal bundle E over \(B^*\times X\), \(B^*\) the irreducible connections. By the slant product with \(c_ 2(E)\) one obtains, after passing to the adjoint map, a map \(\mu\) : \(H^ 2(X;{\mathbb{Z}})\to H^ 2(B^*;{\mathbb{Z}})\). If M(g) were compact one could take the Kronecker product with the fundamental homology class [M(g)] in \(H_ 2(B^*;{\mathbb{Z}})\) from the author’s paper [J. Differ. Geom. 26, 397-428 (1987)] to obtain \(\Gamma\) (g,w). In general he identifies a correction term associated with the description of the end of M(g), which will give a class in \(H^ 2(X;{\mathbb{Z}})\) when M(g) is not compact. This is the content of Proposition 2.9. The proof of the main theorem mentioned above (2.15) is finished by studying the variation of g and \(\omega\).

In general it is at the moment very difficult to compute the Donaldson invariant but if X happens to be an algebraic surface it is closely related to stable vector bundles over it. This is carried out in part (a) of section III. In part (b) he applies this to the Dolgachev surfaces \(D_{p,q}\) which are obtained from the elliptic surface Y diffeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) by logarithmic transformation of multiplicity p and q in two nonsingular fibres. If \((p,q)=1\) these surfaces are homeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) by Freedman’s classification of 1-connected 4-manifolds [M. Freedman, ibid. 17, 357-453 (1982; Zbl 0528.57011)]. But the author uses his invariant combined with information about stable vector bundles to show that \(D_{2,3}\) is not diffeomorphic to \({\mathbb{C}}P^ 2\#9{\mathbb{C}}P^ 2\) (Theorem 3.24). (These computations were later on generalized by R. Friedman and J. Morgan [ibid. 27, 297-369 (1988)], C. Okonek and A. Van de Ven [Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)], and others.)

In section IV (Appendix) the author indicates how his results extend to take account of fundamental groups. As an application he proves (Cor. 4.2) that there is no self-diffeomorphism f on the Godeaux surface S (the orbit space of the quintic in \({\mathbb{C}}P^ 3\) by the obvious \({\mathbb{Z}}_ 5\)-action) such that \(f^*K=K\otimes L^ i\), \(i=1,2,3,4\). Here K is the canonical line bundle and L the flat complex line bundle associated with the covering. (In an article by I. Hambleton and the reviewer [Math. Ann. 280, 85-104 (1988; Zbl 0616.57009)] it was later on shown that these maps do exist as homeomorphisms.)

Reviewer: M.Kreck

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57R55 | Differentiable structures in differential topology |