RELATIVE K-CYCLES AND LOCAL ELLIPTIC BOUNDARY-CONDITIONS

In this paper, we discuss the following conjecture raised by Baum and Douglas: For any first order elliptic differential operator D on a smo oth manifold nil with boundary partial derivative M, D possesses a (lo cal) elliptic boundary condition if and only if partial derivative[D] = 0 in K-1(partial derivative M), where [D] is the relative K-cycle in K-0(M, partial derivative M) corresponding to D. We prove the ''if'' part of this conjecture for dim(M) not equal 4, 5. 6, 7 and the ''only if'' part of the conjecture for arbitrary dimension.