v 1 [ m at h . D G ] 2 5 M ar 1 99 9 Real embeddings and the Atiyah - Patodi - Singer index theorem for Dirac operators ∗

We present the details of our embedding proof, which was announced in [DZ1], of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [APS1]. Introduction The index theorem of Atiyah, Patodi and Singer [APS1, (4.3)] for Dirac operators on manifolds with boundary has played important roles in various problems in geometry, topology as well as mathematical physics. Not surprisingly then, there are by now quite a number of proofs of this index theorem other than Atiyah, Patodi and Singer’s original proof [APS1]. Among these proofs we mention those of Cheeger [C1, 2] (see also Chou [Ch]), Bismut-Cheeger [BC1] and Melrose [M]. One common point underlying all these proofs (including the original one) is that they can all be viewed, in one way or another, as certain extensions to manifolds with boundary of the heat kernel proof of the local index theorem for Dirac operators on closed manifolds (cf. [BeGV]). That is, one starts with a Mckean-Singer type formula and then studies the small time asymptotics of the corresponding heat kernels. In particular, one makes use of the explicit formulas for the heat kernel of the Laplace operators on the cylinder ([APS1], [M]) and/or cone ([BC1], [C1, 2], [Ch]) (being attached the boundary) for the analysis near the boundary. The η-invariant on the boundary, which was first defined in [APS1], appears naturally during the process. Now recall that Atiyah and Singer [AS] also have a K-theoretic proof of their index theorem for elliptic operators on closed manifolds. In such a proof, one Partially supported by NSF grant DMS-9022140 when both authors were visiting MSRI in 1994. Partially supported by NSF and Alfred P. Sloan Foundation. Partially supported by the CNNSF, EMC and the Qiu Shi Foundation. 1 transforms the problem, through direct image constructions in K-theory, to a sphere and then applies the Bott periodicity theorem on the sphere to establish the result. It is thus natural to ask whether the strategy of Atiyah-Singer’s Ktheoretic ideas can be used to prove the Atiyah-Patodi-Singer index theorem for manifolds with boundary. The purpose of this paper is to present such a proof, of which an announcement of basic ideas has already appeared in [DZ1]. Briefly speaking, we embed the manifold with boundary under consideration into a ball, instead of a sphere, so that it maps the boundary of the original manifold to the boundary sphere of the ball, and reduce the problem to the ball. Now since any vector bundle on the ball is topologically trivial, one obtains the result immediately. This works even when the original manifold has no boundary, giving a proof of the Atiyah-Singer index theorem for Dirac operators. The Bott periodicity theorem is thus not needed. Observe that in [AS], Atiyah and Singer made heavy use of the techniques of pseudodifferential operators, which is not suitable for treating directly the global elliptic boundary problems. This is the first serious difficulty in extending directly the arguments in [AS] to deal with the Atiyah-Patodi-Singer boundary problems. On the other hand, Bismut and Lebeau developed in [BL] a general and direct localization procedure which applies to a wide range of localization problems involving Dirac type operators. For example, it has lead to a direct analytic treatment of the index theorem for Dirac operators on closed manifolds along the lines of [AS] (cf. [Z, Remark 2.6]), as well as a localization formula for η-invariants of Dirac operators [BZ] which may be viewed as an odd dimensional analogue of the main result in [BL]. It is these techniques and results that will be used in the present paper, giving an embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [APS1]. In the proof described in [DZ1], we also used in an essential way Cheeger’s cone method [C1, 2]. The reason being, in order to apply Bismut-Lebeau’s method [BL], we need to transfer the Atiyah-Patodi-Singer boundary problem to an elliptic problem on certain manifolds with cone-like singularity. Now, in the present paper, we will show that how one can avoid the analysis on the cone at all. This is done by considering the Atiyah-Patodi-Singer type boundary value problem for certain non-differetial operators arising naturally from the analysis in [BL]. In this way, one no longer encounters the heat kernel analysis on cylinders and/or cones which are essential for the other proofs of the Atiyah-Patodi-Singer index theorem. We regard this as a major technical simplification with respect to [DZ1]. In a separate paper [DZ3], we will further extend the main result of this paper to the case of families. In particular, we will give a new proof of the family index theorem of Bismut-Cheeger [BC1, 2] and Melrose-Piazza [MP] along the lines of this paper. See also the book of Lawson-Michelsohn [LM] for a comprehensive treatment of this approach.