In this paper, the author generalizes two index theorems to relative index theorems on partitioned manifolds.
In the introduction, the author first posts Roe’s index theorem on partitioned manifolds: let \(D\) be the Dirac operator over a complete Riemannian manifold and \(M\) is partitioned by a submanifold \(N\) of codimension 1 into two submanifolds \(M^+\) and \(M^-\) with common boundary \(N=M^+\cap M^-=\partial M^+=\partial M^-\). With Roe’s cyclic 1-cocycle \(\zeta_N\), he proves \[ (\zeta_N)_*(\operatorname{c-ind}(D))=-\frac{1}{8\pi i}\operatorname{index}(D_N^+), \] where \(\operatorname{index}(D^+_N)\) is the Fredholm index of \(D^+_N\).
Then, the author points out that when \(\operatorname{dim}(M)\) is even, this identity is zero, therefore not interesting. Hence, in a previous paper, the author constructs a coarse Toeplitz index \(\operatorname{c-ind(\phi,D)}\) and Toeplitz operator \(T_{\phi|_N}\), and proves the following theorem. \[ (\zeta_N)_*(\operatorname{c-ind}(\phi,D))=-\frac{1}{8\pi i}\operatorname{index}(T_{\phi|_N}), \] for an even dimensional \(M\).
In the main part of the paper, the author starts to construct relative indices and proves two theorems to relative index environments.
Theorem 4.4 \[ (\zeta_*)(\operatorname{c-ind}(D_1,D_2))=-\frac{1}{8\pi i}\operatorname{ind}_t(D_{N_1},D_{N_2}). \] This theorem generalizes Roe’s theorem and it also gives a new proof of well definedness of the relative topological index \(\operatorname{ind}_t(D_{N_1},D_{N_2})\). Also this theorem has the strong connection with a result of M. Karami et al. [Bull. Sci. Math. 153, 57–71 (2019; Zbl 1444.58006)].
Theorem 5.1 \[ (\zeta_*)(\operatorname{c-ind}(\phi_1,D_1,\phi_2,D_2))=-\frac{1}{8\pi i}\operatorname{ind}_t(\phi_{N_1},D_{N_1},\phi_{N_2},D_{N_2}). \] This is the counterpart of Theorem 4.4 and it is also a generalization of the author’s previous result.
The proofs of theorems both contain two steps: reduce the general case to product case and prove the product case.