Relative k-cycles and local elliptic boundary conditions

Abstract

In this paper, we discuss the following conjecture raised by Baum and Douglas: For any first order elliptic differential operator D on a smooth manifold M with boundary ∂M D possesses a (local) elliptic boundary condition if and only if ∂[D]=0 in K1(∂M), where [D] is the relative K-cycle in Ko(M,∂M) corresponding to D. We prove the “if” part of this conjecture for dim(M)≠4,5,6,7 and the “only if” part of the conjecture for arbitrary dimension.