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In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp. the 2-sphere S2) has the Wecken property for n-valued maps for all n ∈ ℕ (resp. all n ≥ 3). In the case n = 2 and S2, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split n-valued map  Open image in new window  of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q: X̂ → X with a subset of the coordinate maps of a lift of the n-valued split map  Open image in new window .

Fixed points of n-valued maps on surfaces and the Wecken property—a configuration space approach