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Index defects in the theory of spectral boundary value problems. (English) Zbl 1072.58014
Gil, Juan (ed.) et al., Aspects of boundary problems in analysis and geometry. Basel: Birkhäuser (ISBN 3-7643-7069-6/hbk). Operator Theory: Advances and Applications 151, 170-238 (2004).
The index of elliptic operators on closed manifolds is a homotopy invariant of the principal symbol. This survey treats homotopy invariants of elliptic operators on manifolds with boundary, related to the Fredholm index.
Let \(M\) be a compact manifold with boundary, and \(D\) a first-order elliptic differential operator. The authors assume that in a collar neighborhood of the boundary \(\partial M\), \(D\) takes the form \[ D=\frac{\partial}{\partial t}+A, \] where \(A\), the so-called tangential operator, is a self-adjoint operator on \(\partial M\).
There exists a topological obstruction to the existence of Fredholm classical boundary value problems, i.e., with local boundary condition, for the operator \(D\). This obstruction is non-zero for many operators of interest, like the Dirac operator, or the Hirzebruch signature operator. By the work of M. F. Atiyah, V. K. Patodi and I. M. Singer [Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975; Zbl 0297.58008)], one can construct nevertheless non-local (i.e., pseudodifferential) boundary conditions which define Fredholm operators. The Atiyah-Patodi-Singer formula famously computes the index of \(D\) with such a non-local condition as the integral of a locally-defined density in the interior, minus half of the eta-invariant of the tangential operator \(A\).
The authors remark that neither the index nor these two contributions are homotopy invariant. Thus they are led to introduce the notion of index defect, which is a quantity depending only on the operator near the boundary, whose sum with the index is homotopy-invariant.
The spectral flow is an obstruction to the existence of index defects on the set of all elliptic operators. The authors restrict themselves to subspaces of operators on which the spectral flow (viewed as a singular \(1\)-cochain) is exact.
They describe such index defects in three cases: when \(A\) is odd and \(\dim(M)\) is even, when \(A\) is even and \(\dim(M)\) is odd (here \(A\) must be pseudodifferential), and when \(\partial M\) is a \(n\)-sheeted covering (in which case they treat the \(\pmod n\) index). The parity refers to the behaviour of the principal symbol of the tangential operator \(A\) under the antipodal map. In all these three settings, the spectral flow condition is fulfilled.
The results appear with full proofs in other papers of the authors, see for instance A. Savin and B. Sternin [Adv. Math. 82, No. 2, 173–203 (2004; Zbl 1043.58012)].
For the entire collection see [Zbl 1050.58002].
58J20 Index theory and related fixed-point theorems on manifolds
58J32 Boundary value problems on manifolds
58J28 Eta-invariants, Chern-Simons invariants
46L85 Noncommutative topology