Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps

Details Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps

1998-12-09

arxiv.org/abs/math/9812060v3

Mathematics

Differential Geometry

86 pages. Simplifications and corrections incorporated. Title streamlined

Scientific paper

The present article is the first in a series whose ultimate goal is to prove the Kotschick-Morgan conjecture concerning the wall-crossing formula for the Donaldson invariants of a four-manifold with b^+ = 1. The conjecture asserts that the wall-crossing terms due to changes in the metric depend at most on the homotopy type of the four-manifold and the degree of the invariant. Our principal interest in this conjecture is due to the fact that its proof is expected to resemble that of an important intermediate step towards a proof of Witten's conjecture concerning the relation between Donaldson and Seiberg-Witten invariants (hep-th/9411102, hep-th/9709193), using PU(2) monopoles as described in (dg-ga/9709022, dg-ga/9712005). Moreover, it affords us another venue in which to address some of the technical difficulties arising in our work on Witten's conjecture. The additional difficulties in the case of PU(2) monopoles are due to the more complicated gluing theory, the presence of obstructions to deformation and gluing, and the need to consider links of positive-dimensional families of `reducibles' even in the presence of `simple type' assumptions. Witten's conjecture should then follow from a final step analogous to Goettsche's computation of the wall-crossing terms, assuming that the Kotschick-Morgan conjecture holds (alg-geom/9506018).

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