Relative $K$-cycles and elliptic boundary conditions

In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator D on smooth manifold M with boundary $\partial M$, D possesses an elliptic boundary condition if and only if $\partial [D] = 0$ in ${K_1}(\partial M)$, where [D] is the relative K-cycle in ${K_0}(M,\partial M)$ corresponding to D. We prove the "if" part of this conjecture for $dim(M) \ne 4,5,6,7$ and the "only if" part of the conjecture for arbitrary dimension.