Relative ^-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 dM , D possesses an elliptic boundary condition if and only if d[D] = 0 in Kx(dM), where [D] is the relative AT-cycle in K0(M, dM) corresponding to D . We prove the "if part of this conjecture for dim(M) ± 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 dM. We always suppose that M is embedded in some compact smooth manifold M without boundary of the same dimension (e.g., o M can be taken as double of M). We denote M= M\dM. Furthermore, we assume that Eq and Ex (in fact, all the vector bundles in this paper) are smooth complex Hermitian vector bundles over M and that D : C°°(Eo) —> C°°(EX) is a first-order elliptic differential operator from smooth sections of Eq to that of Ex . By HS(M, E¡) and Hs(dM, E¡) we shall denote the Sobolev spaces of sections of E¡ and E¿\dM with respect to fixed smooth measures on M and dM, 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] e K0(M, dM) S KK(C0(M),C) (here C0(M) is the algebra of continuous functions on M which vanish on dM) corresponding to D (see [2-4] for details). From the definition of relative K-homology group Kq(M, dM) given by Baum, Douglas, and Taylor, the boundary map d : KQ(M, dM) —> A^i (dM) is very concrete [2-4]. Also Baum and Douglas conjectured that the only obstruction for D possessing elliptic boundary conditions is that d[D] ^ 0. More precisely, the following conjecture first appeared in [2] in a closely related form. Conjecture. There exist a vector bundle E2 over dM and a zeroth-order pseudodifferential operator B defined from Cx(dM,Eo) to Cco(dM, E2) suchthat (A) = « H°(M,EX) (M,E0)^ © Hxl2(dM,E2) Received by the editors March 24, 1992 and, in revised form, June 25, 1992. 1991 Mathematics Subject Classification. Primary 46L80, 46M20, 19K33, 35S15, 35G15. ©1993 American Mathematical Society 0273-0979/93 $1.00+$.25 per page