  In this paper, the first of two, we introduce an alternative definition of
the Chang--Weinberger--Yu relative higher index, which is thought of as a
relative analogue of the Mishchenko--Fomenko index pairing. A main result of
this paper is that our map coincides with the existing relative higher index
maps. We make use of this fact for understanding the relative higher index.
First, we relate the relative higher index with the higher index of amalgamated
free product groups. Second, we define the dual relative higher index map and
show its rational surjectivity under certain assumptions.
