Nielsen coincidence, fixed point and root theories of n-valued maps

Abstract

Let (φ, ψ) be an (m, n)-valued pair of maps φ, ψ : X Y , where φ is an m-valued map and ψ is n-valued, 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, thenwhere < 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 point number of Helga Schirmer to the Nielsen root number of Michael Brown.

On the Nielsen root theory of n-valued maps

“…By Theorem 3.1 of [4], since φ is n-valued and of degree d, then it is homotopic to φ n,d . By Proposition 5.1 of [6], N (φ n,d : 1) = |d| and therefore N (φ : 1) = |d| by Proposition 2.1 above. Since φ n,d has |d| roots at 1, then ψ = φ n,d is the required n-valued map.…”

“…There is only one compact connected 1-manifold without boundary: the circle S 1 . Although, for (n, m)-valued pairs of self-maps of the circle S 1 , Corollary 5.1 of [6] proves that the Nielsen coincidence number is a sharp lower bound for the number of coincidences of all maps (n, m)-homotopic to them, that does not imply that Conjecture 4.2 is true for the Nielsen root number of an n-valued self-map of the circle. As we pointed out in Sect.…”

“…The restrictions of Φ and Ψ to X × {t} for 0 ≤ t ≤ 1 are denoted by φ t and ψ t , respectively. By Lemma 2.1 of [6], a coincidence class of the (n, m)-valued pair (φ t , ψ t ) is contained in a unique coincidence class of the (n, m)-valued pair (Φ, Ψ).…”

“…The Nielsen coincidence number for (n, m)-valued pairs of maps is homotopy invariant by Theorem 2.1 of [6]. That is, let Φ, Ψ : X × I Y be an (n, m)-valued pair of homotopies, then N (φ 0 : ψ 0 ) = N (φ 1 : ψ 1 ).…”

“…Nielsen root theory for single-valued maps was first formalized in [2], but important results concerning roots had been established much earlier by Hopf in [9]. The Nielsen root theory of n-valued maps was introduced in [6] as a tool for studying the Nielsen coincidence theory of such maps. In the present paper, the focus is on the Nielsen root theory of n-valued maps for its own sake.…”