The orientation of Yang-Mills moduli spaces and 4-manifold topology.(English)Zbl 0683.57005

This important paper has two purposes. The first is to generalize the author’s previous results concerning intersection forms of closed smooth 4-manifolds. The second is the construction of a canonical orientation of the Yang-Mills moduli space. Besides its application to the present paper, this is a fundamental information which is used in various other applications of gauge theory to 4-manifolds.
The results concerning the intersection form are the following. Theorem 1: If X is a closed, oriented smooth 4-manifold whose intersection form Q: $$H^ 2(X;{\mathbb{Z}})/Torsion\to {\mathbb{Z}}$$ is negative definite, then the form is equivalent over the integers to the standard form $$(-1)\oplus (- 1)\oplus...\oplus (-1).$$ Theorem 2: Let X be a closed, oriented smooth 4- manifold with the following three properties: (i) $$H_ 1(X;{\mathbb{Z}})$$ has no 2-torsion. (ii) The intersection form Q on $$H^ 2(X)/Torsion$$ has a positive part of rank 1 or 2. (iii) The intersection form is even. Then Q is equivalent over the integers to a hyperbolic form of rank 2 or 4.
Theorem 1 generalizes the author’s corresponding result for 1-connected manifolds [ibid. 18, 279-315 (1983; Zbl 0507.57010)], Theorem 2 was also proved previously by the author for 1-connected manifolds [ibid. 24, 275- 341 (1986; Zbl 0635.57007)]. The methods of the proofs are similar to those in the earlier mentioned papers and based on a detailed analysis of the moduli space of anti-self-dual (ASD) connection on certain SU(2) bundles over X. These moduli spaces are not compact but can be compactified by adding strata involving the moduli spaces coming from SU(2) bundles with smaller second Chern class. If X is not simply connected, the lowest stratum is more complicated creating new difficulties. These are resolved by means of a perturbation argument. This is the content of section 2.
The construction of the canonical orientation on the moduli space of Yang-Mills instantons (ASD connections) is given in section 3. The methods come from index theory for differential operators. The orientation depends on a homology orientation of X. The choice of the canonical orientation is motivated by the case where X is a complex Kähler surface, where the ASD connections may be identified with certain holomorphic vector bundles and their moduli space has a complex structure.
The final section 4 contains among other applications a discussion of another proof of Theorem 1 following ideas of R. Fintushel and R. Stern [ibid. 20, 523-539 (1984; Zbl 0562.53023)].
Reviewer: M.Kreck

MSC:

 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R80 $$h$$- and $$s$$-cobordism 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 81T08 Constructive quantum field theory
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