Loading...
Please wait, while we are loading the content...
Similar Documents
Homotopy equivalence and Donaldson invariants when b^+ = 1. I: Cobordisms of moduli spaces and continuity of gluing maps
| Content Provider | Semantic Scholar |
|---|---|
| Author | Feehan, Paul M. N. Leness, Thomas G. |
| Copyright Year | 1998 |
| Abstract | 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, in essence, 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 (i) 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 [63], using PU(2) monopoles as described in [20], [17], and (ii) 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 considerably 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. Consequently, it is extremely useful to examine the simpler case of the Kotschick-Morgan conjecture in parallel and separately take into account the difficulties that arise in the case of Witten's conjecture. Witten's conjecture should follow from a final step analogous to Göttsche's computation of the wall-crossing terms, assuming that the Kotschick-Morgan conjecture holds [27]. In this article we present the proof of the first half of a general gluing theorem endowed with several important enhancements not present in previous treatments due to Taubes, Donaldson, and Mrowka. We prove that the Uhlenbeck-compactified moduli spaces of instantons on S 4 for two nearby metrics, which a priori need only be cobordant, are actually homeomorphic and diffeomorphic on smooth strata. We also show that the gluing maps, with obstructions to deformation permitted, extend to maps on the closure of the gluing data which are continuous with respect to the Uhlenbeck topology. The second half of the proof of our general gluing theorem is proved in the sequel [14]. We note that Kronheimer and Mrowka have conjectured a relation between the Yang-Mills and Seiberg-Witten Floer homologies for certain three-manifolds Y [37], [39], [48] and they have a recently announced a program to prove this conjecture using moduli spaces of PU(2) monopoles over cylinders Y × R. One of the difficulties in their program, … |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/9812060v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/9812060v2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
