SO(3)-monopoles: The overlap problem

TL;DR

This paper discusses the technical challenges in the SO(3)-monopole program related to computing intersection numbers of links in moduli space cobordisms, and introduces methods to address these difficulties.

Contribution

It provides a framework for computing intersection numbers involving multiple strata in the links of singularities within the SO(3)-monopole cobordism.

Findings

Develops methods for intersection number computations in unions of open sets.

Addresses the complexity of multiple strata in link boundaries.

Lays groundwork for further analysis in the SO(3)-monopole program.

Abstract

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.