AN INDEX THEOREM FOR FAMILIES OF DIRAC OPERATORS ON ODD-DIMENSIONAL MANIFOLDS WITH BOUNDARY

For a family of Dirac operators, acting on Hermitian Clifford modules over the odd-dimensional compact manifolds with boundary which are the fibres of a fibration with compact base, we compute the Chern charact er of the index, in K-1 of the base. Although we assume a product deco mposition near the boundary, we make no assumptions on invertibility o f the boundary family and instead obtain a family of self-adjoint Fred holm operators by choice of an auxiliary family of projections respect ing the Z(2) decomposition of bundles over the boundary. In case the b oundary family is invertible, this projection can be taken to be the A tiyah-Patodi-Singer projection and the resulting formula is as conject ured by Bismut and Cheeger. The derivation of the index formula is eff ected by the combination of the superconnection formalism of Quillen a nd Bismut, the calculus of b-pseudodifferential operators and suspensi on.