 
Summary: Commun. Math. Phys. 182, 303318 (1996) Communications I
Mathematical
Physics
© SpringerVerlag 1996
Splitting of the Family Index
Xianzhe Dai1
, Weiping Zhang2
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA.
Email: xdai@math.usc.edu
2
Nankai Institute of Mathematics, Tianjin 300071, P.R. China.
Received: 29 January 1996/Accepted: 23 April 1996
Abstract: We establish a general splitting formula for index bundles of families of
Dirac type operators. Among the applications, our result provides a positive answer
to a question of Bismut and Cheeger [BC2].
Introduction
Due to the increasing influence of the topological quantum field theory (cf. [A]),
it becomes very important to study the behavior of natural analytic invariants un
der the splitting of manifolds. The first splitting formula, for the most fundamen
tal invariantsthe index of Dirac operators, was proved by AtiyahPatodiSinger
