Inoue–Kabaya Chain Map

Abstract

In this chapter, we study a chain map introduced by Inoue and Kabaya, and observe its properties and applications. Roughly speaking, the chain map plays a key role to bridge the quandle homology and relative group homology. In Sect. 8.1, we give a review of the original definition of the map, and describe an outline to address the chain map in detail. Next, in Sect. 8.2, we demonstrate a philosophy to bridge between the map and topological applications (in particular, the fundamental 3-class). After that, following the outline and philosophy, we describe concrete applications. To be precise, in Sect. 8.3, we recover the Chern-Simons invariants of hyperbolic links, and in Sect. 8.4, we will reconsider the bilinear cohomology pairings of links.