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Real embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operators. (English) Zbl 0995.58014

The authors provide a proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary [M. F. Atiyah, V. K. Patodi, and 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 [X. Dai and 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.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J28 Eta-invariants, Chern-Simons invariants
58J30 Spectral flows
58J32 Boundary value problems on manifolds
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