## Nielsen numbers of n-valued fiber maps (2008)

Venue: | Journal of Fixed Point Theory and Applications |

Citations: | 2 - 2 self |

### BibTeX

@ARTICLE{Brown08nielsennumbers,

author = {Robert F. Brown},

title = {Nielsen numbers of n-valued fiber maps},

journal = {Journal of Fixed Point Theory and Applications},

year = {2008},

pages = {183--201}

}

### OpenURL

### Abstract

The Nielsen number for n-valued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of n-valued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of n-valued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of single-valued maps. If the fibration is orientable, the product formula for single-valued fiber maps fails to generalize, but a “semi-product formula ” is obtained. In this way, the class of n-valued multimaps for which the Nielsen number can be computed is substantially enlarged. Subject Classification 55M20, 54C60 1

### Citations

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Citation Context ... unit circle in the complex plane and c+ and c− are the intersections of S1 with the closed upper and lower halfplanes, respectively. Since c+ and c− are contractible, by Corollary 11.6 on page 53 of =-=[20]-=- there are fiber-preserving homeomorphisms h+ : c+ × Y → p−1 (c+) and h− : c− × Y → p−1 (c−). Let c+ ∩ c− = {z0, z1} = S 0 and orient c+ and c− from z0 to z1. For ɛ = +, − and v = 0, 1, define homeomo... |

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Citation Context ...X × I, write δ(x, t) = {b1, b2, . . . , bn} and let U1, U2, . . . , Un be n disjoint open contractible subsets of B, such that Uj ∩ δ(x, t) = bj for each j = 1, 2, . . . , n. By Corollary 62.8.15 of =-=[19]-=-, there are homotopy equivalences ζj : p−1 (Uj) → Uj ×Y with πUjζj = p and θj : Uj × Y → p−1 (Uj) with pθj = πUj . Since δ is upper semi-continuous with respect to the product topology on X × I, there... |

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Citation Context ...and p ′ : E ′ → B ′ be fibrations. A pair of maps f : E → E ′, ¯ f : B → B ′ such that p ′ f = ¯ fp is a morphism of fibrations ([6], p. 390), more commonly called a fiber map f with induced map ¯ f (=-=[14]-=-, p. 75). If ¯ f(b) = b then f takes the fiber p−1 (b) to itself and the restriction of f to the fiber is denoted by fb : p−1 (b) → p−1 (b). We extend the class of fiber maps to the setting of n-value... |

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Citation Context ...space, then it has the unique path lifting property, that is, if ¯ω, ¯ω ′ ∈ E I such that p¯ω(t) = p¯ω ′ (t) for all t ∈ I and ¯ω(t0) = ¯ω ′ (t0) for some t0 ∈ I, then ¯ω(t) = ¯ω ′ (t) for all t ∈ I (=-=[10]-=-, Prop. 1.34, p. 62). Generalizing Theorem 2.1 of [4], we have Proposition 2.2. Let ∆: X × I ⊸ Y be an n-valued multimap and define φ, ψ : X ⊸ Y by φ(x) = ∆(x, 0) and ψ(x) = ∆(x, 1). If φ is w-split a... |

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Citation Context ...0) = f(x) and pF (x, t) = H(x, t) for all (x, t) ∈ X × I. Let p: E → B and p ′ : E ′ → B ′ be fibrations. A pair of maps f : E → E ′, ¯ f : B → B ′ such that p ′ f = ¯ fp is a morphism of fibrations (=-=[6]-=-, p. 390), more commonly called a fiber map f with induced map ¯ f ([14], p. 75). If ¯ f(b) = b then f takes the fiber p−1 (b) to itself and the restriction of f to the fiber is denoted by fb : p−1 (b... |

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Citation Context ...lued function Ψ: S 1 × Y ⊸ S 1 × Y as follows: for (b, y) ∈ S 1 × Y and e = θ(b, y), if Φ(e) = {e1, e2, . . . , en}, then Ψ(b, y) = {ζ(e1), ζ(e2), . . . , ζ(en)}. By Theorems 1 and 1 ′ on page 113 of =-=[3]-=-, Ψ is continuous, so Ψ is an n-valued fiber map and its induced multimap is φ. For v = 0, 1, let gbv : {bv}×Y → {bv}×Y be the restriction of Ψ. Let θbv : {bv} × Y → p −1 (bv) and ζbv : p −1 (bv) → {b... |

10 |
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Citation Context ...the fixed points of the approximation. The Nielsen number N(φ) is the number of essential fixed point classes, that is, those of nonzero index. The following result is a consequence of Theorem 4.1 of =-=[12]-=-. 1 Proposition 4.1. Let p: E → S 1 be a fibration and let f : E → E be a fiber map with induced map ¯ f : S 1 → S 1 of degree d. If d = 1, then N(f) = 0. Otherwise, let b1, b2, . . . , b|1−d| be poin... |

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Citation Context ...is a consequence of the corresponding property of Φ in the same manner. 3 Fix-finiteness The main result, Theorem 6, of [16], a generalization of a classical result for single-valued maps due to Hopf =-=[13]-=-, states that an n-valued multimap φ: X ⊸ X on a finite polyhedron can be approximated arbitrarily closely by an n-valued multimap with only finitely many fixed points, each of them in a maximal simpl... |

7 |
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(Show Context)
Citation Context ..., for each x ∈ X, is an unordered subset of n points of Y and φ is both upper and lower semi-continuous. Schirmer introduced the Nielsen fixed point theory of n-valued multimaps in a series of papers =-=[16]-=-, [17], [18]. The main result, there called the “minimum theorem”([18], Theorem 5.2) states that if φ: X ⊸ X is an n-valued multimap of a compact triangulated manifold of dimension at least 3, then φ ... |

6 |
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(Show Context)
Citation Context ... Nielsen number N(Φ) to the calculation of Nielsen numbers of single-valued functions. (The class of single-valued maps for which the Nielsen number can be calculated is quite large; see for instance =-=[9]-=- and [11].) In Section 5, we consider orientable fibrations over the circle. Nielsen numbers of single-valued fiber maps of such fibrations satisfy a product formula that does not hold in general for ... |

6 |
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(Show Context)
Citation Context ...e identity map for all [ω] ∈ π1(B, b0). The definition is independent of the choice of the regular lifting function that determines τ. The following product formula is a consequence of Theorem 5.6 of =-=[21]-=-. Proposition 5.1. Let p: E → S1 be an orientable fibration. If f : E → E is a fiber map with induced map ¯ f : S1 → S1 , then N(f) = N( ¯ f)N(fb). When we consider n-valued fiber maps of such fibrati... |

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(Show Context)
Citation Context ..., ψ2(x), . . . , ψn(x)} for all x ∈ X. If ψ is an n-split n-valued multimap so that the ψj are single-valued maps, then ψ is just called a split n-valued multimap. The following classical result from =-=[2]-=- is an important tool in the study of n-valued multimaps. Lemma 2.1. (Splitting Lemma) Let φ: X ⊸ Y be an n-valued multimap and let Γφ = {(x, y) ∈ X × Y : y ∈ φ(x)} be the graph of φ. Then πX : Γφ → X... |

5 |
An index and Nielsen number for n-valued multifunctions
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(Show Context)
Citation Context ...each x ∈ X, is an unordered subset of n points of Y and φ is both upper and lower semi-continuous. Schirmer introduced the Nielsen fixed point theory of n-valued multimaps in a series of papers [16], =-=[17]-=-, [18]. The main result, there called the “minimum theorem”([18], Theorem 5.2) states that if φ: X ⊸ X is an n-valued multimap of a compact triangulated manifold of dimension at least 3, then φ is n-v... |

4 |
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(Show Context)
Citation Context ...U) × I by T = (p −1 (U) × {0}) ∪ (p −1 (U \ V ) × I) ∪ (p −1 (b) × I). Define H : T → E by { H(e, t) = Φ1(e) if t = 0 or e ∈ p−1 (U\V ) h(e, t) if e ∈ p−1 (b) By the fiber homotopy extension theorem (=-=[1]-=-, Theorem 2.2), we can extend H to a fiber-preserving homotopy H : p −1 (U) × I → E. Define Γ1 : p −1 (U) → E by Γ1(e) = H(e, 1). Noting that Γ1(e) = Φ1(e) for e ∈ p −1 (U \ V ), we define Ψ: E ⊸ E by... |

4 | Fixed points of n-valued multimaps of the circle
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(Show Context)
Citation Context ...papers were not concerned with the calculation of the Nielsen number. There are only two examples in those papers, both are 2-valued multimaps of the circle, for which the Nielsen number is given. In =-=[4]-=-, in addition to extending the minimum theorem to nvalued multimaps of the circle, we determined the Nielsen numbers for all the n-valued multimaps of the circle as follows. We defined the degree of a... |

4 | A minimum theorem for n-valued multifunctions - Schirmer - 1985 |

2 | The Lefschetz number of an n-valued multimap - Brown - 2007 |

1 |
On fiber homotopy equivalence
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(Show Context)
Citation Context ...+(z, t) if z ∈ c+ h ′ −(z, t) if z ∈ c− which is a well-defined fiber map because h ′ −(zv, y) = h+(zv, y) for v = 0, 1. Since the restriction of h+ to z × Y for any z ∈ c+ is a homotopy equivalence, =-=[7]-=- implies that h is a fiber homotopy equivalence. Lemma 5.1. Let p: E → S 1 be an orientable fibration and let Φ: E ⊸ E be an n-valued fiber map. Let b0 and b1 be fixed points of the induced multimap φ... |

1 |
The equivalence of fiber spaces and bundles
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(Show Context)
Citation Context ... are fiber homotopic to the identity maps. 15Theorem 5.1. An orientable fibration p: E → S 1 with fiber Y = p −1 (b0) is fiber homotopy equivalent to π S 1 : S 1 × Y → S 1 . Proof. By the theorem of =-=[8]-=-, the fibration is fiber homotopy equivalent to a bundle. Since orientability is preserved by fiber homotopy equivalence, we may assume without loss of generality that p: E → B is an orientable bundle... |