Paper detail

SO(3)-monopoles: The overlap problem

The SO(3)-monopole program, initiated by Pidstrigatch and Tyurin [arXiv:dg-ga/9507004], yields a relationship between the Donaldson and Seiberg-Witten invariants through a cobordism between the moduli spaces defining these invariants. The main technical difficulty in this program lies in describing the links of singularities in this cobordism arising from the Seiberg-Witten moduli subspaces. In related articles, we defined maps which, essentially, define normal bundles of strata of these singularities. The link in question is then the boundary of the union of the tubular neighborhoods associated with these normal bundles. However, the SO(3)-monopole program requires the computation of intersection numbers with links where more than one stratum appears in the family of singularities and thus more than one tubular neighborhood appears in the definition of the link. Computations of intersection numbers in unions of open sets have proved difficult for even two open sets, as early work of Leness [arXiv:dg-ga/9603016] and Ozsvath [1994] demonstrated. In this note, we give a brief introduction to our monograph [arXiv:math/0203047], in which we implement these computations.