Appendix C Index of Parameter-Dependent Elliptic Families

In this appendix, we discuss the index theorem for parameter-dependent elliptic families due to Nistor (2003).

Let D(v) : C∞(M,E) −→ C∞(M,F ) be a parameter-dependent elliptic family with parameters v ∈ V , where V is a finite-dimensional vector space. Recall that the principal symbol

σ(D) : π∗E −→ π∗F, π : (T ∗M ⊕ V ) \ 0 −→ M,

of a parameter-dependent elliptic family (in the sense of Agranovich-Vishik) is invertible.