Cobordism invariance and the well-definedness of local index

In the previous papers, Furuta, Yoshida and the author gave a definition of analytic index theory of Dirac-type operator on open manifolds by making use of some geometric structure on an open covering of the end of the open manifold and a perturbation of the Dirac-type operator. In this paper, we show the cobordism invariance of the index, and as an application we show the well-definedness of the index with respect to the choice of the open covering.