%0 Journal Article %T Relative $K$-cycles and elliptic boundary conditions %A Guihua Gong %J Mathematics %D 1993 %I arXiv %X 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. %U http://arxiv.org/abs/math/9301213v1