PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees (Q2744703)

This is the second of a series of two papers, which follows [ibid., 57-133 (2001; Zbl 0983.57024), see above]. NEWLINENEWLINENEWLINEIn these two papers the authors prove Witten's conjecture relating the Donaldson and Seiberg-Witten series, for terms of degree less than or equal to \(c-2\), where \(c=-(7\chi+11\sigma)/4\) and \(\chi\) and \(\sigma\) are the Euler characteristic and signature of the four-manifold. The proof applies to four-manifolds \(X\) with \(b_{1}=0\), \(b^{+}>1\) and which are abundant, SW-simple type and effective. NEWLINENEWLINENEWLINEAfter the work carried out in the companion paper, this paper continues by studying the orientation of PU(2) monopole moduli spaces (depending on a homology orientation of the four-manifold \(X\)), the induced orientation on the link of the moduli space of anti-self-dual connections (the Donaldson strata) inside this moduli space, and the orientation of the link of the SW-strata. NEWLINENEWLINENEWLINEThen the authors define the classes \(\mu(\alpha)\) on the PU(2) monopole moduli space for homology classes \(\alpha\) on the four-manifold, and construct geometric representatives [as in \textit{P.B. Kronheimer} and \textit{T. S. Mrowka}, J. Differ. Geom. 41, No. 3, 573-734 (1995; Zbl 0842.57022)], with appropriate transversality properties with respect to the Donaldson strata. There is a new class (the determinantal class) which is a representative of the Poincaré dual of the hyperplane section in the link of the top level Donaldson stratum. NEWLINENEWLINENEWLINENow take an intersection \(Z\) of geometric representatives of the \(\mu\) classes and determinantal classes, with dimension \(1\). Then \(Z\) intersects the link of the Donaldson stratum in a finite number of points directly related to the Donaldson invariant of \(X\). The transversality properties of the geometric representatives make that the intersection only occurs in the top level of the link of the Donaldson strata. NEWLINENEWLINENEWLINEThe analysis of the restriction of the \(\mu\) classes to the link of the SW-strata in the top level (when the monopoles do not have any bubbling), allows to make cohomological computations about the number of points of the intersection of \(Z\) with the link of the top level SW-strata, by using the computation of the Chern classes of the virtual normal bundle of the SW-strata in the companion paper. Note that in principle there is no reason not to expect that \(Z\) does not intersect lower level SW-strata. NEWLINENEWLINENEWLINEExplicit computations are carried out under the simplifying assumption \(b_{1}(X)=0\). The results relate the Donaldson and Seiberg-Witten series, with the appearance of hyper-geometric functions in accordance with the conjecture in [\textit{G. Moore} and \textit{E. Witten}, Adv. Theor. Math. Phys. 1, No. 2, 298-387 (1997; Zbl 0899.57021)]. This type of functions appear in the Donaldson invariants of four-manifolds which are (potentially) not of simple type [\textit{V. Muñoz}, Donaldson invariants of non-simple type four-manifolds, Topology, in press]. NEWLINENEWLINENEWLINEA discussion of the blow-up procedure (and how the formula relating Donaldson and Seiberg-Witten invariants changes) is given, since it is necessary to avoid reducible anti-self-dual connections (see [\textit{J. W. Morgan} and \textit{T. S. Mrowka}, Int. Math. Res. Not. 1992, No. 10, 223-230 (1992; Zbl 0787.57011)] for the original idea). NEWLINENEWLINENEWLINEFinally the condition of SW-simple type is added to simplify furthermore the formulas already obtained. NEWLINENEWLINENEWLINEThe fact that only top-level SW-strata are dealt with in this series of two papers imposes the restriction that the equality between the Donaldson and Seiberg-Witten series is through degrees less than or equal to \(c-2\).