Boundary value problems on manifolds with fibered boundary. (English) Zbl 1078.58014
Summary: We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the \(C^*\)-algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah-Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given.
MSC:
| 58J32 | Boundary value problems on manifolds |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 19K56 | Index theory |
Keywords:
boundary value problems; fibered boundary; Atiyah-Patodi-Singer theory; Atiyah-Bott obstruction; K-theoryReferences:
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