Relative 𝐾-cycles and elliptic boundary conditions

Abstract

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 ∂ M \partial M , D possesses an elliptic boundary condition if and only if ∂ [ D ] = 0 \partial [D] = 0 in K 1 ( ∂ M ) {K_1}(\partial M) , where [D] is the relative K-cycle in K 0 ( M , ∂ M ) {K_0}(M,\partial M) corresponding to D. We prove the "if" part of this conjecture for d i m ( M ) ≠ 4 , 5 , 6 , 7 dim(M) \ne 4,5,6,7 and the "only if" part of the conjecture for arbitrary dimension.