We give a framework of localization for the index of a Dirac-type operator on
an open manifold. Suppose the open manifold has a compact subset whose
complement is covered by a family of finitely many open subsets, each of which
has a structure of the total space of a torus bundle. Under an acyclic
condition we define the index of the Dirac-type operator by using the
Witten-type deformation, and show that the index has several properties, such
as excision property and a product formula. In particular, we show that the
index is localized on the compact set.