The index defect in the theory of nonlocal problems and the \(\eta\)-invariant.

*(English. Russian original)*Zbl 1083.58022
Sb. Math. 195, No. 9, 1321-1358 (2004); translations from Mat. Sb. 195, No. 9, 85-126 (2004).

Let \(M\) be a compact manifold with boundary, where the boundary \(Y\) is a covering space of a manifold \(X\). Using the covering map, one can define, for any vector bundle on \(Y\), a direct image, which is a vector bundle on \(X\) the space of whose sections is naturally isomorphic to the space of sections of the original vector bundle on \(Y\). This construction provides a way to define, for a differential operator on \(M\), a boundary-value problem involving classical boundary conditions on \(X\).

In this paper the authors provide a thorough discussion of the index theory of such boundary-value problems (which the authors call non-local because each point in \(X\) corresponds to several points in \(Y\)). In particular the Shapiro-Lopatinskii conditions define an elliptic non-local problem, which the authors prove is Fredholm. When the boundary covering map is the quotient map for a free action of a finite group, the authors prove an index theorem, identifying the analytic index with a number (modulo the number of sheets in the cover) determined by a symbol class in an appropriate \(K\)-theory group. A local index formula is deferred to a later paper. The proof of the index theorem is modeled on the Atiyah-Singer embedding proof, although the embedding is in an \(N\)-universal bundle (\(N\) sufficiently large) for the group acting on the boundary.

Among the highlights of this version of index theory are the following. When \(M\) is a compact oriented Riemannian \(4k\)-manifold and \(Y\) admits an orientation-reversing smooth involution without fixed points, the Hirzebruch signature operator admits an elliptic non-local boundary condition with Fredholm index equal to the signature of \(M\) [see C.-C. Hsiung, J. Differ. Geom. 6, 595–598 (1971; Zbl 0248.58008)]. For trivial (product) covers, the paper’s index theorem is that of D. S. Freed and R. B. Melrose [Invent. Math. 107, 283–299 (1992; Zbl 0760.58039)]. The relative eta invariant of M. F. Atiyah, V. K. Patodi, and I. M. Singer [Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976; Zbl 0325.58015)] appears in the defect formula relating the index of an operator with Atiyah-Patodi-Singer boundary conditions to the index with non-local boundary conditions. The authors discuss PoincarĂ© duality involving the \(K\)-theory of a \(C^{*}\)-algebra that generalizes one class of examples considered by J. Rosenberg [Geom. Dedicata 100, 65–84 (2003; Zbl 1036.58020)]. Similar ideas are discussed for more general fibered boundaries in [A. Savin, \(K\)-Theory 34, 71–98 (2005; Zbl 1087.58013)].

In this paper the authors provide a thorough discussion of the index theory of such boundary-value problems (which the authors call non-local because each point in \(X\) corresponds to several points in \(Y\)). In particular the Shapiro-Lopatinskii conditions define an elliptic non-local problem, which the authors prove is Fredholm. When the boundary covering map is the quotient map for a free action of a finite group, the authors prove an index theorem, identifying the analytic index with a number (modulo the number of sheets in the cover) determined by a symbol class in an appropriate \(K\)-theory group. A local index formula is deferred to a later paper. The proof of the index theorem is modeled on the Atiyah-Singer embedding proof, although the embedding is in an \(N\)-universal bundle (\(N\) sufficiently large) for the group acting on the boundary.

Among the highlights of this version of index theory are the following. When \(M\) is a compact oriented Riemannian \(4k\)-manifold and \(Y\) admits an orientation-reversing smooth involution without fixed points, the Hirzebruch signature operator admits an elliptic non-local boundary condition with Fredholm index equal to the signature of \(M\) [see C.-C. Hsiung, J. Differ. Geom. 6, 595–598 (1971; Zbl 0248.58008)]. For trivial (product) covers, the paper’s index theorem is that of D. S. Freed and R. B. Melrose [Invent. Math. 107, 283–299 (1992; Zbl 0760.58039)]. The relative eta invariant of M. F. Atiyah, V. K. Patodi, and I. M. Singer [Math. Proc. Camb. Philos. Soc. 79, 71–99 (1976; Zbl 0325.58015)] appears in the defect formula relating the index of an operator with Atiyah-Patodi-Singer boundary conditions to the index with non-local boundary conditions. The authors discuss PoincarĂ© duality involving the \(K\)-theory of a \(C^{*}\)-algebra that generalizes one class of examples considered by J. Rosenberg [Geom. Dedicata 100, 65–84 (2003; Zbl 1036.58020)]. Similar ideas are discussed for more general fibered boundaries in [A. Savin, \(K\)-Theory 34, 71–98 (2005; Zbl 1087.58013)].

Reviewer: Peter Haskell (Blacksburg)