Self-adjoint local boundary problems on compact surfaces. II. Family index

Abstract

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems. We address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X . We compute the K 1 .X/-valued index in terms of the topological data over the boundary. The second result is the universality of the index: we show that the index is a universal additive homotopy invariant for such families if the vanishing on families of invertible operators is assumed. PrefaceAn index theory for families of elliptic operators on a closed manifold was developed by Atiyah and Singer in [5]. For a family of such operators, parametrized by points of a compact space X, the K 0 .X /-valued analytical index was computed there in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by Atiyah, Patodi, and Singer in [3]; the index of a family in this case takes values in the K 1 group of a base space.If a manifold has a non-empty boundary, the situation becomes more complicated. The integer-valued index of a single boundary value problem was computed by Atiyah and Bott [2] and Boutet de Monvel [7]. This result was generalized to the K 0 .X /-valued index for families of boundary value problems by Melo, Schick, and Schrohe in [17].