# A cyclic cocycle and relative index theorems on partitioned manifolds

## TL;DR

This paper extends Roe's cyclic 1-cocycle to relative settings and proves two generalized relative index theorems for partitioned manifolds, broadening the scope of index theory in geometric analysis.

## Contribution

It introduces a relative cyclic cocycle and establishes new relative index theorems for partitioned manifolds, generalizing existing results.

## Key findings

- Extended Roe's cyclic 1-cocycle to relative contexts
- Proved two new relative index theorems for partitioned manifolds
- Generalized previous index theorems, including a variant of a known result

## Abstract

In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of [M. Karami-A.H.S. Sadegh-M.E. Zadeh, arXiv:1411.6090, Theorem 3.3].