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Families of Dirac operators, boundaries and the b-calculus

Richard B. Melrose tex2html_wrap_inline16 and Paolo Piazza tex2html_wrap_inline18


A version of the Atiyah-Patodi-Singer index theorem is proved for general families of Dirac operators on compact manifolds with boundary. The vanishing of the analytic index of the boundary family, in tex2html_wrap_inline24 of the base, allows us to define, through an explicit trivialization, a smooth family of boundary conditions of generalized Atiyah-Patodi-Singer type. The calculus of b-pseudodifferential operators is then employed to establish the family index formula. A relative index formula, describing the effect of changing the choice of the trivialization, is also given. In case the boundary family is invertible the form of the index theorem obtained by Bismut and Cheeger is recovered.

Richard B. Melrose
Sat Mar 30 08:52:12 EST 1996