Abstract
A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral bound- ary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodierential opera- tors on the fibres (with boundary) of a fibration; a version of Poincare duality is also shown in this setting.
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