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 Theorem 3.3 in “Relative-partitioned index theorem” by M. Karami, A.H.S. Sadegh and M.E. Zadeh.
"A Cyclic Cocycle and Relative Index Theorems on Partitioned Manifolds." Tokyo J. Math. 42 (2) 431 - 448, December 2019. https://doi.org/10.3836/tjm/1502179288