  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.
