Relative K-cycles and Elliptic Boundary Conditions

In this paper, we discuss the following conjecture raised by BaumDouglas: For any first-order elliptic differential operator D on smooth manifold M with boundary ∂M , D possesses an elliptic boundary condition if and only if ∂[D] = 0 in K1(∂M), where [D] is the relative K-cycle in K0(M, ∂M) corresponding to D. We prove the “if” part of this conjecture for dim(M) 6= 4, 5, 6, 7 and the “only if” part of the conjecture for arbitrary dimension. First we fix some notation. M is a compact oriented smooth manifold with smooth boundary ∂M . We always suppose that M is embedded in some compact smooth manifold M̃ without boundary of the same dimension (e.g., M̃ can be taken as double of M). We denote ◦ M= M \ ∂M . Furthermore, we assume that E0 and E1 (in fact, all the vector bundles in this paper) are smooth complex Hermitian vector bundles over M and that D : C∞(E0) → C ∞(E1) is a first-order elliptic differential operator from smooth sections of E0 to that of E1. By H (M,Ei) and H(∂M,Ei) we shall denote the Sobolev spaces of sections of Ei and Ei|∂M with respect to fixed smooth measures on M and ∂M , respectively. The elliptic boundary value problem (an elliptic operator with an elliptic boundary condition) has been studied for a long time. As noted in [1, 5, 6] and other references, there exist topological obstructions to impose an elliptic boundary condition on the above D. A fundamental problem is to find all such obstructions. Baum, Douglas, and Taylor constructed a relative K-cycle [D] ∈ K0(M,∂M) ∼= KK(C0( ◦ M),C) (here C0( ◦ M) is the algebra of continuous functions on M which vanish on ∂M) corresponding to D (see [2–4] for details). From the definition of relative K-homology group K0(M,∂M) given by Baum, Douglas, and Taylor, the boundary map ∂ : K0(M,∂M) −→ K1(∂M) is very concrete [2–4]. Also Baum and Douglas conjectured that the only obstruction for D possessing elliptic boundary conditions is that ∂[D] 6= 0. More precisely, the following conjecture first appeared in [2] in a closely related form. Conjecture. There exist a vector bundle E2 over ∂M and a zeroth-order pseudoReceived by the editors March 24, 1992 and, in revised form, June 25, 1992. 1991 Mathematics Subject Classification. Primary 46L80, 46M20, 19K33, 35S15, 35G15. c ©1993 American Mathematical Society 0273-0979/93 $1.00 + $.25 per page