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We study the coincidence theory of maps between two manifolds of the samedimension from an axiomatic viewpoint. First we look at coincidences of mapsbetween manifolds where one of the maps is orientation true, and give a set ofaxioms such that characterizes the local index (which is an integer valuedfunction). Then we consider coincidence theory for arbitrary pairs of mapsbetween two manifolds. Similarly we provide a set of axioms which characterizethe 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 ofmaps between two manifolds we provide a set of axioms which charaterize thelocal Reidemeister trace which is an element of an abelian group which dependson the pair of functions. These results extend known results for coincidencesbetween orientable differentiable manifolds.

Axioms for the coincidence index of maps between manifolds of the same dimension