  We study the coincidence theory of maps between two manifolds of the same
dimension from an axiomatic viewpoint. First we look at coincidences of maps
between manifolds where one of the maps is orientation true, and give a set of
axioms such that characterizes the local index (which is an integer valued
function). Then we consider coincidence theory for arbitrary pairs of maps
between two manifolds. Similarly we provide a set of axioms which characterize
the local index, which in this case is a function with values in $\Z\oplus
\Z_2$. We also show in each setting that the group of values for the index
(either $\Z$ or $\Z\oplus \Z_2$) is determined by the axioms.
  Finally, for the general case of coincidence theory for arbitrary pairs of
maps between two manifolds we provide a set of axioms which charaterize the
local Reidemeister trace which is an element of an abelian group which depends
on the pair of functions. These results extend known results for coincidences
between orientable differentiable manifolds.
