Real embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operators (Q5947036)

The authors provide a proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [\textit{M. F. Atiyah}, \textit{V. K. Patodi}, and \textit{I. M. Singer}, Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)] that is based on embedding a manifold with boundary in a ball. Unlike other proofs, including the proof announced in [\textit{X. Dai} and \textit{W. Zhang}, C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1293--1297 (1994; Zbl 0817.58040)], this proof uses heat kernel analysis on neither cones nor cylinders. Ingredients of the proof include: introduction of a vector-bundle map on the ball that is invertible off the embedded submanifold; trivialization of the domain and range bundles on the ball; and analytically proven localization and variation formulas. Thus the authors' proof can be regarded as the analogue, for the Atiyah-Patodi-Singer theorem, of the proof of the Atiyah-Singer index theorem inspired by Grothendieck's Riemann-Roch theorem.