Localized ASD moduli spaces based on the reduced cohomology group over Casson handles

In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with respect to the standard Sobolev spaces. We apply it to construct moduli theory mainly over Casson handles, and obtain another approach to see high complexity of their smooth structure.