  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 $\p M$, $D$ 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 $\dim(M)$ $\not=$ 4, 5, 6, 7 and the ``only if''
part of the conjecture for arbitrary dimension.
