A cyclic cocycle and relative index theorems on partitioned manifolds. (English) Zbl 1444.19009

Seto, Tatsuki

Tokyo J. Math. 42, No. 2, 431-448 (2019).

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.

Reviewer: Lin Shan (San Juan)

MSC:

19K56	Index theory
46L87	Noncommutative differential geometry
58J20	Index theory and related fixed-point theorems on manifolds

Keywords:

coarse geometry; Dirac operator; index theory; partitioned manifolds; relative index theorem; Roe cocycle; Toeplitz operator

Citations:

Zbl 1444.58006

References:

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