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COINCIDENCES OF PROJECTIONS AND LINEAR nVALUED MAPS OF TORI
, 2009
"... We prove that the Nielsen fixed point number N(ϕ) of an nvalued map ϕ: X ⊸ X of a compact connected triangulated orientable qmanifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of ϕ, a subset of X×X, to the two factors. For certain q ×q integer matr ..."
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We prove that the Nielsen fixed point number N(ϕ) of an nvalued map ϕ: X ⊸ X of a compact connected triangulated orientable qmanifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of ϕ, a subset of X×X, to the two factors. For certain q ×q integer matrices A, there exist “linear ” nvalued maps Φn,A,σ: T q ⊸ T q of qtori that generalize the singlevalued maps fA: T q → T q induced by the linear transformations TA: R q → R q defined by TA(v) = Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear nvalued maps. Subject Classification 55M20, 54C60 1
NIELSEN COINCIDENCE, FIXED POINT AND ROOT THEORIES OF nVALUED MAPS
, 2013
"... Let (φ, ψ) be an (m, n)valued pair of maps φ, ψ: X ⊸ Y, where φ is an mvalued map and ψ is nvalued, on connected finite polyhedra. A point x ∈ X is a coincidence point of φ and ψ if φ(x) ∩ ψ(x) = ∅. We define a Nielsen coincidence number N(φ: ψ) which is a lower bound for the number of coincide ..."
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Let (φ, ψ) be an (m, n)valued pair of maps φ, ψ: X ⊸ Y, where φ is an mvalued map and ψ is nvalued, on connected finite polyhedra. A point x ∈ X is a coincidence point of φ and ψ if φ(x) ∩ ψ(x) = ∅. We define a Nielsen coincidence number N(φ: ψ) which is a lower bound for the number of coincidence points of all (m, n)valued pairs of maps homotopic to (φ, ψ). We calculate N(φ: ψ) for all (m, n)valued pairs of maps of the circle and show that N(φ: ψ) is a sharp lower bound in that setting. Specifically, if φ is of degree a and ψ of degree b, then N(φ: ψ) = an − bm < m, n>, where < m, n> is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for nvalued maps of compact connected Lie groups that relate the Nielsen fixed