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The active bijection for oriented matroids (and real hyperplane arrangements,and graphs, as particular cases) is introduced and investigated by the authorsin a series of papers. Given any oriented matroid defined on a linearly orderedground set, we exhibit one particularit\'e of its bases, which we call itsactive basis, with remarkable properties. It preserves activities (for orientedmatroids in the sense of Las Vergnas, for matroid bases in the sense of Tutte),as well as some active partitions of the ground set associated with orientedmatroids and matroid bases. It yields a canonical bijection between classes ofreorientations and bases [...]. It also yields a refined bijection between allreorientations and subsets of the ground set. Those bijections are related tovarious Tutte polynomial expressions [...]. They contain various noticeablebijections involving orientations/signatures/reorientations and spanningtrees/simplices/bases of a graph/real hyperplane arrangement/oriented matroid.[...]  In previous papers of this series, we defined the active bijection betweenbounded regions and uniactive internal bases by means of fully optimal bases(No. 1), and we defined a decomposition of activities for matroid bases bymeans of [...] particular sequences of minors (companion paper, No. 2.a). Thepresent paper is central in the series. First, we define a decomposition ofactivities for oriented matroids, using the same sequences of minors, yieldinga decomposition of an oriented matroid into bounded regions of minors. Second,we use the previous results together to provide the canonical and refinedactive bijections alluded to above. We also give an overview and examples ofthe various results of independent interest involved in the construction. Theyarise as soon as the ground set of an oriented matroid is linearly ordered.

The Active Bijection 2.b - Decomposition of activities for oriented matroids, and general definitions of the active bijection