Splitting of the family index (Q679377)

The authors prove a splitting formula for a family of Dirac operators acting on a closed manifold. More precisely, let \(Z\) be splitted by a closed hypersurface \(Y\) (which may vary with the parameter \(b\in B\)) into two pieces \(Z_1\) and \(Z_2\), and let \(P_1\) and \(P_2\) be two spectral sections over \(Y\) encoding the respective boundary conditions. Then the index bundle \(\text{ind }D_Z\) in \(K^{\dim Z}(B)\) may be splitted as follows: \[ \text{ind }D_Z= \text{ind}(D_{Z_1},P_1)+ \text{ind}(D_{Z_2},1- P_2)+ [P_1- p_2]. \] The corresponding cohomological additivity property of the Chern character of the index bundle has been obtained before by \textit{J.-M. Bismut} and \textit{J. Cheeger} [Differential geometry, Pitman Monogr. Surv. Pure Appl. Math. 52, 59-83 (1991; Zbl 0727.58045)]. As a by-product of the proof they also obtain a relative index theorem for different spectral sections \(P\) and \(Q\): \[ \text{ind}(D_{Z_1},P)- \text{ind}(D_{Z_2},Q)= [Q- P]. \] {}.