APPEARED IN BULLETIN OF THE

arXiv:math/9301213v1 [math.AP] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 104-108RELATIVE K-CYCLES AND ELLIPTIC BOUNDARYCONDITIONSGUIHUA GONGDedicated to Professor Zejian Jiang on his seventieth birthdayAbstract. In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator D on smooth manifoldM with boundary ∂M, D possesses an elliptic boundary condition if and onlyif ∂[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) ̸=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 withsmooth boundary ∂M.

We always suppose that M is embedded in some compactsmooth manifold fM without boundary of the same dimension (e.g., fM can be takenas double of M). We denote◦M= M \ ∂M.

Furthermore, we assume that E0 andE1 (in fact, all the vector bundles in this paper) are smooth complex Hermitianvector bundles over M and that D : C∞(E0) →C∞(E1) is a first-order ellipticdifferential operator from smooth sections of E0 to that of E1. By Hs(M, Ei) andHs(∂M, Ei) we shall denote the Sobolev spaces of sections of Ei and Ei|∂M withrespect to fixed smooth measures on M and ∂M, respectively.The elliptic boundary value problem (an elliptic operator with an elliptic bound-ary condition) has been studied for a long time.

As noted in [1, 5, 6] and otherreferences, there exist topological obstructions to impose an elliptic boundary con-dition 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 whichvanish on ∂M) corresponding to D (see [2–4] for details). From the definition ofrelative K-homology group K0(M, ∂M) given by Baum, Douglas, and Taylor, theboundary map ∂: K0(M, ∂M) −→K1(∂M) is very concrete [2–4].

Also Baum andDouglas conjectured that the only obstruction for D possessing elliptic boundaryconditions is that ∂[D] ̸= 0. More precisely, the following conjecture first appearedin [2] in a closely related form.Conjecture.

There exist a vector bundle E2 over ∂M and a zeroth-order pseudo-Received by the editors March 24, 1992 and, in revised form, June 25, 1992.1991 Mathematics Subject Classification. Primary 46L80, 46M20, 19K33, 35S15, 35G15.c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2GUIHUA GONGdifferential operator B defined from C∞(∂M, E0) to C∞(∂M, E2) such thatDB ◦γ:H1(M, E0) −→H0(M, E1)⊕H1/2(∂M, E2)is Fredholm if and only if ∂[D] = 0 in K1(∂M), where γ : H1(M, E0) −→H1/2(∂M, E0)is the trace map.Remark 1. Let D be as above with pincipal symbol p(x, ξ).

A zeroth-order pseudo-differential operator B with principal symbol b(x, ξ) from C∞(∂M, E0) to C∞(∂M, E2)is said to be elliptic to D (see [6], p. 233) if, for every x ∈∂M and ξ ∈T ∗x(∂M), themap M +x,ξ ∋u −→b(x, ξ)u(0) ∈(E2)x is bijective, where T ∗x(∂M) and (E2)x arethe fibres at the point x of the cotangent bundle T ∗∂M and the bundle E2, and,furthermore, M +x,ξ is the set of all u ∈C∞(R, (E0)t) with p(x, ξ −i ddt · nx)u(t) = 0(nx is the interior conormal vector of M at x) which are bounded on R+. If B is el-liptic to D, then the aboveDB◦γis Fredholm.

Such a systemDB◦γis often calledan elliptic boundary value problem or an elliptic operator with an elliptic boundarycondition; meanwhile, D is also said to possess an elliptic boundary condition.Remark 2. Although the above elliptic boundary condition is used in most refer-ences, the original form of the conjecture in [2] is in a slightly different form fromthe above.

In [2] the operator for the boundary condition is of the form γ ◦B,where B is a zeroth-order pseudo-differential operator from E0 to a smooth vectorbundle over a neighborhood of M. The reason we use a slightly different form ofthe conjecture is as follows: for general zeroth-order pseudo-differential operatorB defined on fM, there is no canonical way to restrict B to M as an operatorBM : Hs(M) −→Hs(M) when s > 0. So one needs to put some restriction on B.One of the natural restrictions is that B has the transmission property with respectto ∂M (see [5]).

We also prove our theorem for this kind of boundary condition (seeTheorem 1). It must be pointed out that the existence of a boundary condition oftype B ◦γ implies the existence of that of type γ ◦B.In this paper, we prove the “only if” part of the conjecture which can be thoughtof as a generalization of Corollary 4.2 in [4] (there B is a differential operator).Conversely, we prove that if dim(M) ̸= 4, 5, 6, 7 and ∂[D] = 0, then D possesses anelliptic boundary condition as in Remark 1.

Hence the “if” part of the conjecturehas been proved for dim(M) ̸= 4, 5, 6, 7. The cases of dim(M) being equal to 4, 5, 6,or 7 are still open, but we prove a theorem which can be thought of as the “if” partof the conjecture in the sense of stablization in K-homology group for arbitrarydimension.

Our results will be useful for constructing absolute K-cycles in K0(M)which are preimages of [D] ∈K0(M, ∂M) under the canonical map from K0(M) toK0(M, ∂M) when ∂[D] = 0.Our main results are the following:Theorem 1. (“only if ” part) ∂[D] = 0 if one of the following is true:(i) There exist a smooth vector bundle E2 over ∂M and a zeroth-order pseudo-differential operator B from E0|∂M to E2 such thatDB◦γin the conjecture isFredholm.

RELATIVE K-CYCLES AND ELLIPTIC BOUNDARY CONDITIONS3(ii) There exist a bundle E2 over a neighborhood of M in fM and a zeroth-orderpseudo-differential operatorB with transmission property with respect to ∂M fromE0 to E2 such thatDγ ◦B:H1(M, E0) −→H0(M, E1)⊕H1/2(∂M, E2)is Fredholm.Theorem 2. (“if ” part) If ∂[D] = 0, then there exists a first-order elliptic dif-ferential operator D1 acting on smooth vector bundles over M with [D1] = 0 inK0(M, ∂M) such that D ⊕D1 possesses an elliptic boundary condition as in Re-mark 1, and, furthermore, if dim(M) ̸= 4, 5, 6, 7, then D itself possesses an ellipticboundary condition.The main idea of the proof of Theorem 1 is to construct an intertwining between∂[D] and a trivial element in K1(∂M).

In the proof, we use Calderon projection,functional calculus of pseudo-differential operators (including Boutet de Monveltype operators), and the techniques in the proof of Proposition 4.5 of [4].The proof of Theorem 2 makes use of two key lemmas (see below).Let ST ∗∂M be the unit sphere bundle of T ∗∂M over ∂M and π : ST ∗∂M −→∂M be the canonical projection map. Let eE0 = π∗(E0|∂M) be the bundle overST ∗∂M.

We write the principal symbol of D, in a coordinate neighborhood U ofx ∈∂M, asp(x, xn, ξ, ξn) =n−1Xj=1pj(x, xn)ξj + pn(x, xn)ξn,where xn is the coordinate for the normal direction of ∂M. We defineτ(x, ξ) = ip−1n (x, 0)n−1Xj=1pj(x, 0)ξjfor x ∈∂M and ξ ∈ST ∗∂M.

Then τ(x, ξ) is a map from a fibre of E0 into itself andhas no purely imaginary eigenvalue. Let V± be the subbundle of eE0 over ST ∗∂Mcorresponding to the span of the generalized eigenvectors of τ(x, ξ) correspondingto the eigenvalues with positive/negative real parts.Lemma 1.

∂[D] = 0 if and only if [V+] ∈π∗K0(∂M) ⊂K0(ST ∗∂M).Lemma 2. If E0 and E1 are vector bundles over M which allow a first-order el-liptic differential operator D to act from one to the other, and if dim(M) = n,then(i) f dim(E0) = f dim E1 ≥2[(n−1)/2];(ii) f dim(E0) = f dim E1 ≥2[(n−1)/2]+1 provided n is even and ∂[D] = 0,where f dim denotes dimension of each fibre of the vector bundles.

4GUIHUA GONGThe Proof of Theorem 2. By Lemma 1, if ∂[D] = 0, one has[V+] ∈π∗K0(∂M) ⊂K0(ST ∗∂M).By Lemma 2, f dim(V+) = f dim(E0)2≥2[n/2]−1.

Therefore, f dim(V+) ≥n −1 >dim(ST ∗∂M)2= 2n−32, whenever dim(M) ≥8. Hence there exists a complex vectorbundle E2 over ∂M, such that V+ ∼= π∗E2.

This is also true for dim(M) ≤3,since the collection of complex vector bundles over ST ∗∂M (dim(ST ∗∂M) ≤3)has property of cancellation.Let ψ be the bundle isomorphismV+ψ−→π∗E2↓↓ST ∗∂M−→ST ∗∂MFor any (x, ξ) ∈ST ∗∂M, let b(x, ξ) be the bundle map defined byE0project to−→V+ψ−→E2↓↓ST ∗∂M−→ST ∗∂M−→ST ∗∂MFurthermore, let B : E0|∂M −→E2 be the zeroth-order pseudo-differential op-erator with symbol b(x, ξ). It follows that B is elliptic to D (see [6]).Example 1.

If M is a spinc manifold with smooth boundary and D is the Diracoperator over M, then it is computed in [4] that ∂[D] ̸= 0. Hence D possessesno elliptic boundary condition (even possesses no boundary condition as in theconjecture).Example 2.

For any D, let D∗be the formal adjoint of D.It is easy to provethat [D] = −[D∗] in K0(M, ∂M); hence, ∂[D ⊕D∗] = 0. It follows that D ⊕D∗possesses an elliptic boundary condition provided dim(M) ̸= 4 or 6.

(It should benoted that we only need to exclude the manifolds with dimension 4 and 6 here,since the dimension of the bundle on which D ⊕D∗acts is twice the dimension ofthe bundle on which D acts. )The details of the proofs will appear elsewhere.AcknowledgmentsThis work was done while the author was a postdoctoral fellow at the Universityof Toronto.

He is grateful to Professors George A. Elliott and Man-Deun Choifor their support. The author also thanks Professors Man-Deun Choi, Ronald G.Douglas, George A. Elliott, and Peter Greiner and Ms. Liangqing Li for manyhelpful conversations.

RELATIVE K-CYCLES AND ELLIPTIC BOUNDARY CONDITIONS5References1. M. Atiyah, V. Patodi, and I.

Singer, Spectral asymmetry and riemannian geometry. I, II, III,Math.

Proc. Cambridge Philos.

Soc. 77 (1975), 43–69; 78 (1975), 405–632; 79 (1976), 71–99.2.

P. Baum and R. Douglas, Index theory, bordism, and K-homology, Operator Algebras andK-Theory (R. G. Douglas and C. Schochet, eds. ), Contemp.

Math., vol. 10, Amer.

Math. Soc.,Providence, RI, 1982, pp.

1–31.3., Relative K-homology and C∗-algebra, manuscript.4. P. Baum, R. Douglas, and M. Taylor, Cycles and relative cycles in analytic K-homology, J.Differential Geom.

30 (1989), 761–804.5. L. Boutet de Monvel, Boundary problems for pseudodifferential operators, Acta Math.

126(1971), 11–51.6. L. H¨omander, The analysis of linear partial differential operators.

III, Springer, New York,1985.Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S1A1, and Department of Mathematics, Jilin University, People’s Republic of ChinaCurrent address: Department of Mathematics, Queen’s University, Kingston, Canada K7L3N6E-mail address: guihua@math.toronto.edu

출처: arXiv:9301.213 • 원문 보기