Nielsen type numbers for periodic points of fibre preserving maps

Topology and its Applications 248 (2018) 204–225

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Topology and its Applications www.elsevier.com/locate/topol

Nielsen type numbers for periodic points of fibre preserving maps Changbok Li, Yujin Paek ∗ Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea

a r t i c l e

i n f o

Article history: Received 15 September 2017 Received in revised form 16 March 2018 Accepted 4 July 2018 Available online 15 August 2018 MSC: 55M20 55R05 Keywords: Nielsen type numbers Periodic points Fiber spaces

a b s t r a c t In 1995 Heath, Keppelmann and Wong studied a Nielsen type number NF (f, p) for a fibre preserving map f of fibration p. This number is a lower bound for the least number of fixed points within the fibre homotopy class of f . In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers N PnF (f, p) and N ΦF n (f, p) for periodic points of a fibre preserving map. These numbers, which can be thought of as the dual of periodic Nielsen type numbers for a relative map due to Heath, Schirmer and You, are bigger than the ordinary Nielsen type numbers. The definition of N PnF (f, p) and N ΦF n (f, p) is reminiscent of the naive type addition formulae for periodic points of fibre preserving maps due to Heath and Keppelmann. Some calculations and examples are given. © 2018 Published by Elsevier B.V.

1. Introduction For a self map f : X → X of a compact connected AN R X, a periodic point of f is a point x ∈ X such that there exists a positive number n with f n (x) = x. We denote the set of all periodic points of period dividing n by Φ(f n ) = {x ∈ X|f n (x) = x}. The set of periodic points of least period n will be denoted by Pn (f ) = {x ∈ Φ(f n )|x ∈ / Φ(f m )∀m < n}. For any set S, the symbol #S will denote the cardinality of S. For any natural number n, there are two homotopy invariant Nielsen type numbers N Pn(f ) and N Φn (f ). The first is a lower bound for the number of periodic points of least period n of f , and the second a lower bound for the number of periodic points of all periods m|n of f (see [16,12,15]). The use of fibre techniques has been applied successfully to the computation of the ordinary Nielsen number and Nielsen type numbers for periodic points of a fibre preserving map (see [3,5,7,8,16,20,11,10,1,18,6]). Let p : E → B be a fibration of compact AN R’s, and let f : E → E be a fibre preserving map (i.e., a map for which there exists a map f¯ : B → B with pf = f¯p). In 1995 [11] Heath, Keppelmann and Wong introduced the naïve addition formula expressing the Nielsen number of the fibre preserving map as a simple sum of Nielsen numbers on

* Corresponding author. E-mail address: [email protected] (Y. Paek). https://doi.org/10.1016/j.topol.2018.07.003 0166-8641/© 2018 Published by Elsevier B.V.

C. Li, Y. Paek / Topology and its Applications 248 (2018) 204–225

the fibres, and the Nielsen type number NF (f, p) =



205

N (fb ) of a fibre preserving map f (where ξ is a set

b∈ξ

¯ This number is a lower bound for which contains exactly one point from each essential Nielsen class of f). minimum number, M ΦF (f, p), of fixed points in the fibre homotopy class of the map f , with respect to the fibration p, and NF (f, p) ≥ N (f ). In 2000 [10] Heath and Keppelmann introduced the naïve type addition formulae N Pn (f ) =

 N Pn/per(b) ((f per(b) )b ) and

N Φn (f ) =

b∈ξ



N Φn/per(b) ((f per(b) )b )

b∈ξ

(where ξ is a subset of Φ(f¯n ) which contains one point from each irreducible essential class at level m for all m|n) for the computation of the Nielsen type numbers for periodic points of a fibre preserving map f , based on the naïve addition conditions of [11]. The main objective here then, is to define two new Nielsen type numbers N PnF(f, p) and N ΦF n (f, p) that will provide lower bounds for the numbers M PnF (f, p) = min{#Pn (g) | (g, g¯)  (f, f¯) : p → p} (the minimum numbers of periodic points of period exactly n of maps g that are fibrewise homotopic to f ), and for n M ΦF ¯)  (f, f¯) : p → p} n (f, p) = min{#Φ(g ) | (g, g

(the minimum numbers of periodic points of all periods dividing n of maps g that are fibrewise homotopic to f ), and which uses the naïve type sums for periodic points of a fibre preserving maps from [10] in the same way that [11] uses the naïve sum to define the Nielsen type number NF (f, p). Thus in effect we are restricting our attention to fibre preserving maps and homotopies. An example was given in [10, Example 4.10] which shows that the naïve type addition formulae for periodic points are invalid if N (f¯n ) = 0, and we repeat it here as it will be frequently used, either in this or in modified forms, to illustrate the considerations, results and calculations of this paper. Example 1.1. ([10, 4.10]) Consider the trivial fibration P

T 2 → T 4 = T 2 × T 2 −−−−→ T 2 where p is projection on the second factor. Let f = f0 × f¯ : T 4 → T 4 which is induced by the matrix ⎡

1 ⎢ −1 A=⎣ 0 0

2 0 −1 0 0 0 0 −1

⎤ 0 0⎥ . 1⎦ 1

Thus the induced map f¯ on the base is induced by the 2 × 2 matrix C in the lower right. Since N (f¯) = |det(C − I)| = 1, f¯ has a distinguished fixed point 0 ∈ Φ(f¯), the restriction f0 of f to p−1 (0) is induced by the 2 × 2 matrix B in the upper left. From N (f¯n ) = |det(C n − I)| and N (f n ) = |det(An − I)| = |det(B n − I)||det(C n − I)|, we have that N (f¯6 ) = N (f 6 ) = 0, N Φ6 (f¯) = N P1 (f¯) + N P2 (f¯) + N P3 (f¯) = 1 + 2 + 3 = 6 and N Φ6 (f0 ) = N (f06 ) = 4. Let ξ = {b1 , b2 , b3 , b4 , b5 , b6 } be a set of essential 6-root representatives for f¯ with b1 ∈ Φ(f¯); b2 and b3 = f¯(b2 ) being of least period 2; b4 , b5 = f¯(b4 ) and b6 = f¯2 (b4 ) being of least period 3. Since f¯(ξ) = ξ, then f induces, by restriction, a self map fξ : p−1 (ξ) → p−1 (ξ) on the nonconnected union of fibres. Now f can be regarded as a self map of the pair (T 4 , p−1 (ξ)) and so, according to relative theory in [14], has at least

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206

N P6 (f ; T 4 , p−1 (ξ)) = N P6 (fξ ) =



N P6/per(b) ((f per(b) )b ) =

b∈ξ



N Pm (f¯)N P6/m (f0m ) = 6

m|6

periodic points of least period 6, N Φ6 (f ; T 4 , p−1 (ξ)) = N Φ6 (fξ ) =



N Φ6/per(b) ((f per(b) )b ) = N Φ6 (f0 )N Φ6 (f¯) = 24

b∈ξ

periodic points of period 6 (see Proposition 5.2 and Example 5.6). But N P6 (f ) = 0 and N Φ6 (f ) = 18 (see [10, Example 4.10]). Although for fibre preserving maps f with N (f¯n ) = 0 on nil and solvmanifolds, two naïve type sums coincide with N Pn (f ) and N Φn (f ) respectively, this is in general false. It turns out however that these sums serve as lower bounds the number of periodic points of least period n, and period n within the fibre homotopy class of f . Thus these sums become the important and interesting numbers even when they do ¯ We not coincide with N Pn (f ) and N Φn (f ) respectively. Under a certain condition with the base map f. F shall define the fibrewise Nielsen type number N Pn (f, p) of period n and the fibrewise Nielsen type number  N ΦF N Pn/per(b) ((f per(b) )b ) and n (f, p) for the nth iterate of the fibre preserving map f to be respectively b∈ξ  N Φn/per(b) ((f per(b) )b ), sums over a representative set ξ of periodic points in the base. b∈ξ

From Example 1.1, one might be tempted to guess that our fibrewise Nielsen type numbers for periodic points are simply the relative Nielsen type numbers for periodic points of f from [14], where the subspace involved is taken to be the nonconnected union p−1 (ξ) of the fibres. We will show that our fibrewise Nielsen type numbers coincides with the relative Nielsen type numbers for periodic points of f with respect to the subspace p−1 (ξ) of E. This not only makes the relationship of our work to that of [14] clear, but also helps in some of the proofs. The paper is organized as follows. Following this introduction we give in Section 2 a brief review of the topics we use frequently in this paper. In Section 3, we introduce fibrewise Nielsen type numbers N PnF (f, p) and N ΦF n (f, p), and discuss the properties of these numbers. We show that N PnF (f, p), N ΦF (f, p) are equal to the relative Nielsen type numbers N Pn (f ; E, p−1 (ξ)), N Φn (f ; E, p−1 (ξ)) n −1 −1 F for f : (E, p (ξ)) → (E, p (ξ)), thus N Pn (f, p), N ΦF n (f, p) are lower bounds for the number of periodic points of least period n of f and Φ(f n ), respectively. Both numbers have moreover fibred versions of the usual properties of Nielsen numbers: homotopy invariance, homotopy type invariance and commutativity. n Section 4 discusses the relationship of N PnF (f, p) and N ΦF n (f, p) to each other, and to NF (f , p). We obtain  F F conditions so that N Φn (f, p) can be found as m|n N Pm (f, p), or (by Möbius inversion) so that N PnF (f, p) can be found as a sum of terms which have the form N ΦF k (f, p) for suitable values of k. We also show that n N ΦF (f, p) is greater or equal to N (f , p), the fibrewise Nielsen number of the nth iterate of f , and state F n conditions under which equality holds. We use the results from [9] to obtain conditions so that N ΦF n (f, p)  can be found as sum b∈ξ N ((f n )b ) of ordinary Nielsen numbers on fibres. Finally in Section 5 we give product formulae for the fibre uniform case which hold for N PnF (f, p) and N ΦF n (f, p). All of the results from Section 4 and 5 are used to calculate the two fibrewise Nielsen type numbers N PnF (f, p) and N ΦF n (f, p) in several more sophisticated examples, in particular in certain examples which are obtained from Example 1.1 and its modifications. 2. Review A fairly extensive background is required to appreciate the results of this paper. This includes Nielsen theory, relative Nielsen theory and their extension to the theory of periodic Nielsen classes, and the supporting work involving the Nielsen theory of fibre preserving maps, as well as the theory of nil and solvmanifolds.

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Thus we give a brief review of the concepts from ordinary and relative Nielsen periodic point theory, and theorems for the periodic points of fibre preserving maps from [9,10] that are used frequently. 2.1. Nielsen periodic point theory We will begin by recalling some of the basics of ordinary Nielsen theory and its generalization to the study of periodic points. We use a modified fundamental group approach as in [7,12,15], but we will also need results from [16] to review relative periodic theory in [14]. Suppose that f : X → X is a self map of compact connected AN R X, then ordinary Nielsen theory assigns a non-negative integer N (f ), called the Nielsen number, to f . The Nielsen number counts the number of essential fixed point Nielsen classes of f . The number N (f ) is a homotopy invariant lower bound for min{#(Φ(g)) | g  f } which is sharp in many cases (see for example [2]). For each positive integer n, Φ(f n ) = {x ∈ X : f n (x) = x} denotes the fixed point set of the nth iterate f n of f . For x ∈ Φ(f n ) the period of x, denoted by per(x), is defined to be the smallest positive integer m|n such that x ∈ Φ(f m ). We will denote the set of all geometric Nielsen classes for f n by Φ(f n )/∼. We will use An , B n , or C n to denote generic geometric Nielsen classes of f n . When the n is omitted, then n = 1 is assumed. Nielsen periodic point theory begins from the basic premise that if m|n then Φ(f m ) ⊆ Φ(f n ). This inclusion induces a so called boosting function, γm,n : Φ(f m )/∼→ Φ(f n )/∼, between the Nielsen classes of f m and those of f n . The Nielsen classes can be viewed as arising from a partitioning of the fundamental group into Reidemeister classes defined using the homomorphism induced by f . For this it is necessary to choose a base point x0 ∈ X, along with a path ω from x0 to f (x0 ). Up to canonical bijection the following constructions are independent of these choices. Please note in what follows we shall not distinguish between a path and its path class in the fundamental groupoid of X. The path ω provides “coordinates” for all iterates of f as follows. Define 1ω = ω and for n > 1 let nω = (n − 1)ωf n−1 (ω). Then nω is a path (class) from x0 to f n (x0 ). We abbreviate (nω)−1 by nω −1 . If x0 is a fixed point of f , we will note distinguish between x0 and the constant path at x0 . Note that in this case nx0 = x0 . There is an induced homomorphism f∗nω : π1 (X, x0 ) → π1 (X, x0 ) by f∗nω (α) = nωf∗n (α)nω −1 . The map nω f∗ provides the relation of Reidemeister equivalence on π1 (X, x0 ) by the rule that α ∼ β if and only if there exists δ ∈ π1 (X, x0 ) with α = δβf∗nω (δ −1 ). The Reidemeister class containing α will be denoted by [α]n , and the set of all Reidemeister classes by R(f∗nω ). We call R(f∗nω ) the set of (algebraic) n periodic point classes of f . The symbol R(f n ) denotes the Reidemeister number R(f∗nω ) of f n . The algebraic and geometric Nielsen classes are related by an injective function ρn = ρ : Φ(f n )/∼→ R(f∗nω ) defined as follows. Given x ∈ An we choose a path c from the basepoint x0 to x. We can then define ρ(An ) = [cf n (c−1 )nω −1 ]n . This will be independent of the choice of x within An and c. The set of the geometric Nielsen classes for f n can be regarded as a subset of R(f∗nω ) via ρ and we can define an index for the algebraic Nielsen classes of f n so that the index of [α]n is the same as the index of the geometric class (empty or not) that determines it (see for example [15]). Thus we can define the Nielsen number N (f n) of f n either as the number of essential geometric classes or as the number of essential algebraic classes of f n. A map f is said to be weakly Jiang provided that either N (f ) = 0 or else N (f ) = R(f ). The algebraic boosting function ιm,n : R(f∗mω ) → R(f∗nω ) is defined by

n (n−m)ω ιm,n ([α]m ) = αf∗mω (α)f∗2mω (α) · · · f∗ (α) . The algebraic and the geometric boosting functions are related by the equation ργm,n = ιm,n ρ. We warn the reader that in general the γm,n and their algebraic counterparts ιm,n (see [9]) need not be injective.

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For both algebraic and geometric Nielsen classes at level n we have the notions of reducibility, depth, and length. The map f induces an index preserving bijection on the algebraic or geometric classes at level n in such a way that f∗nω (respectively f n ) is the identity. This allows for a natural breaking up of the algebraic (respectively geometric) classes into orbits in such a way that depth and hence reducibility (or not) are properties of orbits. The orbit of [α]n (An ) is denoted by [α]n  ( An ), respectively. We now define the Nielsen type numbers for periodic points. For each positive integer n we define N Pn (f ) = n#(IEOn ), where IEOn is the set of irreducible essential algebraic orbits of f at level n. Note that the definition, given in terms of orbits, takes into account the fact that if there is one periodic point of least period n then are at least n of them. The other periodic point number, N Φn (f ), is defined by computing the minimal height of a set of n-representatives for f . This is a collection of algebraic orbits at various levels with the property that any essential orbit [α]m  for any m|n reduces to some element in this set. The height of such a set is the sum of the depths of all its members. Theorem 2.1. N Pn (f ) and N Φn (f ) have homotopy invariance, commutativity and homotopy type invariance, and hence N Pn (f ) ≤ M Pn (f ) and N Φn (f ) ≤ M Φn (f ). We now give a brief review of relative Nielsen periodic point theory [13,14]. In 1986 [19] Nielsen fixed point theory was extended to the setting of maps of pairs of spaces, and a relative Nielsen number was introduced which is a homotopy invariant lower bound for the number of fixed points for a map of a pair of compact AN R’s. In 1995 Schirmer’s ideas about relative theory, and the existing ideas about periodic points, were combined to give a common generalization of relative fixed point theory and periodic point theory in the paper [14]. This theory considers the periodic Nielsen theory of a map f : (X, A) → (X, A), of a pair of compact AN R’s with X connected, give two relative Nielsen type number N Pn (f ; X, A) and N Φn (f ; X, A) which are homotopy invariant lower bounds for the numbers of periodic points of least period n and Φ(f n ), respectively. Then A can have more than one path-component, but as A is a compact AN R, the number of its path-components is finite. We denote the path-components of A by Aj , write fA : A → A for the restriction of f to A rather than f¯ as in [14], because we have already used f¯ above, and fj : Aj → Ak for the restriction of f to the component Aj of A rather than f¯j as in [14]. If ν is the inclusion ν : A → X, then ν is a morphism from fA : A → A to f : X → X, that is ν ◦ fA = f ◦ ν. Now let [Aj ] be an fA -cycle of length c(j) (see [13, Definition 3.3]). For each Ak in this cycle there is a commutative diagram f

c(j)

k Ak −−− −→ ⏐ ⏐ ν

Ak ⏐ ⏐ν

(2.1)

f c(j)

X −−−−→ X of path-connected spaces. Note that [13,14] make use both the covering space approach of [16] and the modified fundamental group approach of [12,15]. We assume familiarity with the covering space approach, give here simply an overview of the notation, definitions and results from [14] which we shall use. From [13, Proposition 2.7] we have that, for any l|m, the diagram c(j)

ιl,m

c(j)

(f

c(j)

)∗

c(j)

k F P C((fk )l ) −−−−→ F P C((fk )m ) −−− −−→ F P C((fk )m ) ⏐ ⏐ ⏐ ⏐ν ⏐ν ⏐ν

∗

∗

∗

ιlc(j),n

F P C(f c(j)l ) −−−−−→

F P C(f n )

(f c(j) )∗

−−−−−→

F P C(f n )

(2.2)

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is commutative, where n = mc(j), and F P C(f n ) denotes the set of lifting classes of f n to the universal ˜ weighted by the index [16, I.1.4 and III.1.10]. There exists a bijection χ : F P C(f n ) → R(f nω ) covering X ∗ which is obtained by assigning to each lifting class of F P C(f n ) its coordinate in R(f∗nω ). Note that we are using multiplicative notation unlike [14] which used additive notation even for the non commutative groups. An analogous diagram exists in the fundamental group approach. One should keep in mind that, according c(j) to [13, Section 3], α ¯ is an n-periodic point class of fA if α ¯ ∈ F P C((fk )n/c(j) ) for some component Ak of A. An n-periodic point class α ∈ F P C(f n ) (or R(f∗nω )) of f is called a common n-periodic point class of f and fA if there exists an essential n-periodic point class α ¯ of fA so that ν∗ (¯ α) = α. Furthermore, it is an essential common n-periodic point class of f and fA if it is a common n-periodic point class which is itself essential, and it is an irreducible common n-periodic point class of f and fA if it is common and irreducible. Note that if α is an irreducible common n-periodic point class of f and fA and ν∗ (¯ α) = α, then α ¯ is also irreducible by diagram above. If α ∈ F P C(f n ) (or R(f∗nω )), is respectively, a common, essential common, irreducible common n-periodic point class of f and fA , then f∗ (α) is also, respectively, a common, essential common, irreducible common n-periodic point class of f and fA , hence these properties of an n-periodic point class are properties of the orbit of f n . This fact justifies that we define the first number. Definition 2.2. ([14, 2.3]) The relative Nielsen type number of period n for f : (X, A) → (X, A) is N Pn (f ; X, A) = N Pn (fA ) + N Pn (f ) − N Pn (f, fA ), where N Pn (fA ) is the Nielsen type number of period n for map fA on space A with finitely many path-components (see [13, Definition 3.4]), and N Pn (f, fA ) is defined to be n times the number of common irreducible essential orbits of f n and fAn . Proposition 2.3. ([14, 2.8, 2.9, 2.11]) Every relative map which is homotopic to f : (X, A) → (X, A) has at least N Pn (f ; X, A) periodic points of least period n, N Pn (f ; X, A) ≥ N Pn (fA ) and N Pn (f ; X, A) ≥ N Pn (f ). For the definition of the second number we need to know how the sets of n-representatives of f and fA interact, and hence to extend the definition of a reducible n-periodic orbit given in [13, Definition 3.6] to the relative setting. We will denote the sets of all n-periodic algebraic orbits of f and fA by Orbn (f ) and Orbn (fA ), respectively, and Orbm|n (f ) and Orbm|n (fA ) will denote the sets {Orbm (f ) : m|n} and {Orbm (fA ) : m|n}, respectively, as in [14]. Definition 2.4. ([14, 3.2, 3.3, 3.5]) If m|n, and if α ∈ Orbn (fA )  Orbn (f ) and β ∈ Orbm (fA )  Orbm (f ) are orbits of the maps f : X → X and fA : A → A determined by a map f : (X, A) → (X, A), then α is pseudo-reducible to β if one of the following two conditions is satisfied: (P1) α is either fA -reducible or f -reducible to β, c(j) (P2) α ∈ Orbn (f ), β ∈ Orbm/c(j) (fj ) for some cycle [Aj ], and α is f -reducible to ν∗ ( β) (see diagram (2.2)). A finite subset S of Orbm|n (f )  Orbm|n (fA ) is called a set of n-representatives for f : (X, A) → (X, A) if every essential orbit in Orbm|n (f )  Orbm|n (fA ) is pseudo-reducible to at least one element of S. The relative Nielsen number for the nth iterate of f : (X, A) → (X, A), N Φn (f ; X, A) is defined to be the minimal height of a set of n-representatives for f : (X, A) → (X, A). Proposition 2.5. ([14, 3.7, 3.9, 3.10]) Every relative map which is homotopic to f : (X, A) → (X, A) has at least N Φn (f ; X, A) periodic points of period n, N Φn (f ; X, A) ≥ N Φn (fA ) and N Φn (f ; X, A) ≥ N Φn (f ). 2.2. The naïve type addition formulae for periodic points of fibre preserving maps In this paper we will assume that all of our fibrations F → E → B (with p : E → B) are Hurewicz fibrations. Recall that this means there is a continuous path lifting function

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Λ : Ω = {(ω, e) ∈ B I × E | p(e) = ω(0)} → E I . For any (ω, e) ∈ Ω, Λ(ω, e) is a path in E beginning at e whose projection is ω. For any b ∈ B, Fb will denote p−1 (b). It is in this way that each such ω provides a fibre translation map τω : Fω(0) → Fω(1) by the rule that τω (e) = Λ(ω, e)(1). When ω ∼ ω  then τω ∼ τω . We say that f : E → E is a fibre preserving ¯ map provided it induces a well defined base map f¯ : B → B with pf = f¯p. We will refer to the pair (f, f) ¯ is a fibre preserving map and b ∈ Φ(f¯) then we will use fb (or just f ) as a fibre preserving map. If (f, f) to denote the restriction f |p−1 (b) : Fb → Fb of f to the fibre. In regard to iterates note that if b ∈ Φ(f¯q ) then in Fb the relevant self map becomes (f q )b and (f q )rb = (f qr )b . Note that if per(b) = q > 1 then for any m > 1, (fb )m is undefined because until f is iterated q times, the restriction is not a self map of Fb . In [9] and [10] Heath and Keppelmann gave the fibrewise methods for evaluating the Nielsen type numbers N Φn (f ) and N Pn (f ) for periodic points of self maps f of nilmanifolds and solvmanifolds. In the first [9] they showed for all maps f on nilmanifolds, and for weakly Jiang maps on solvmanifolds, that N Φn (f ) and N Pn (f ) can be computed from the (highly computable) collection of ordinary Nielsen numbers N (f m ) for m|n. For non-weakly Jiang maps this is no longer true. In the second paper [10] they proved naïve type addition formulae for periodic points of fibre preserving maps, which are particularly useful for maps of solvmanifolds that are not weakly Jiang. The idea is that for fibrations that satisfy the so called naïve addition conditions and other conditions (see [10, 3.4]), one can count periodic points in the total space by counting them in the fibres over a representative set of periodic points in the base. We recall these concepts and results from [9] and [10]. A nilmanifold N is a coset space of the form G/Γ where G is a connected simply connected nilpotent Lie group, and Γ is a uniform (i.e., N is compact) discrete subgroup of G. For each nilmanifold N there is a Fadell–Husseini fibration T → N → M with torus fibres and a base which is a nilmanifold of dimension less than that of N . A solvmanifold S is a coset space of the form J/Δ where J is a connected simply connected solvable Lie group, and Δ is a uniform closed (but not necessarily discrete) subgroup of J. Corresponding to each solvmanifold S there is a minimal Mostow fibration M → S → T in which the fibres are nilmanifolds and the base a torus (see [17]). Both the Fadell–Husseini fibration for a nilmanifold N , and the minimal Mostow fibration for a solvmanifold S, have the property that, up to homotopy, every self map on N or S is fibre preserving with respect to these fibrations. This is the basis for the fibre space techniques used to study the periodic classes for these spaces in [9] and [10]. Recall from [9, 2.3], [11, 4.4] that p : E → B satisfies the naïve addition conditions provided that for each p(y) fibre preserving map f of p, and for all y ∈ Φ(f ) belonging to an essential class of f we have that F ixf¯∗ = p∗ (F ixf∗y ) and NK (fp(y) ) = N (fp(y) ), where K is the kernel of the inclusion induced homomorphism from π1 (F ) to π1 (E). All fibre preserving maps on the above fibrations for nil and solvmanifolds satisfy the naïve addition conditions (see [9, 3.3]). The naïve addition conditions allow us to count the ordinary Nielsen number as in the following theorem. Theorem 2.6. ([11, 4.4]) Let p : E → B be a fibration which satisfies the naïve addition conditions for all  fibre preserving maps, and let f be a fibre preserving map inducing f¯ : B → B. Then N (f ) = b∈ξ N (fb ), where ξ ⊆ Φ(f¯) consists of one fixed point from each essential class. For a map f : X → X let ε(f ) denote the collection of essential geometric Nielsen classes of f . A map f is said to be essentially reducible if whenever An ∈ ε(f n ) reduces to level m then its reduction B m is essential. All maps on nil and solvmanifolds are essentially reducible [9, 4.5].

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Theorem 2.7. ([15, 4.2 and 4.6]) For any essentially reducible map f N Φn (f ) =



N Pm (f )

and



N Pn (f ) =

(−1)#τ N Φn:τ (f ),

τ ⊆P(n)

m|n

where P(n) is the set of prime divisors of n and n : τ = n

 p∈τ

p−1 .

The following properties, which do not hold in general, but which do hold for all maps of nil and solvmanifolds (see [9]), are also be needed in order to state and prove the main theorem in [10] and our results. In the presence of essential reducibility a map f is said to have essential torality provided that the following two conditions are met. Firstly we require that the change of level boosting functions are injective on essential classes. In other words, if An = γm,n (B m ) = γm,n (C m ) and An , B m , and C m are all essential, then B m = C m . The second requirement is that the length of an essential orbit of an irreducible class at any level m is equal to m. A map f is said to be essentially reducible to the GCD provided that f is essentially reducible, and whenever a class An ∈ ε(f n ) reduces to both B r and C s (at levels r and s), then there is a class Dq , with q = GCD(r, s), to which both B r and C s reduce. Finally, f has uniqueness of roots for essential classes provided that whenever a class An ∈ ε(f n ) reduces to irreducible classes B r and C s , then r = s and B r = C s . In [9, 4.19] it is shown that uniqueness of roots is actually a consequence of essential reducibility, essentially reducibility to the GCD, and the injectivity portion of essential torality. Proposition 2.8. ([9, 4.5, 4.12, 4.16 and 4.19]) All maps of nil and solvmanifolds are essentially reducible, essentially toral, essentially reducible to the GCD, and have uniqueness of roots for essential classes. In the following M (f, n) denotes the set of maximal divisors m of n for which N (f m ) = 0, IECm (g) denote the set of irreducible essential classes of g m . Theorem 2.9. ([9, 5.1, 5.8]) Let f be essentially toral, essentially reducible to the GCD, and suppose that f m is weakly Jiang for every m ∈ M (f, n). Then N Φn (f ) =



(−1)#μ−1 N (f GCD(μ) ),

∅=μ⊆M (f,n)

where GCD(μ) is the greatest common divisor of all elements of μ. In particular if f n is weakly Jiang and N (f n ) = 0 so that M (f, n) = {n}, then N Φm (f ) = N (f m ) for every m|n. Definition 2.10. ([10, 3.1]) Suppose that g : X → X and n are such that g is essentially reducible, and for all m|n one has that the essential classes at level m have unique roots and N Pm (g) = #(IECm (g)) (i.e., each essential irreducible orbit at level m contains m distinct classes). Then a set of essential n-root representatives of g is a collection ξ ⊆ Φ(g n ) which contains one point from each irreducible essential class at level m for all m|n. From Proposition 2.8 the following theorem applies to all Mostow [solvmanifold] and Fadell–Husseini [nilmanifold] fibrations. Theorem 2.11. ([10, 3.4 and 3.6]) Let (f, f¯) be a fibre preserving map on a fibration p : E → B satisfying the naïve addition conditions. Suppose further that f and f¯ : B → B are essentially reducible, have essential torality, that f¯ is essentially reducible to the GCD, and that f¯n is weakly Jiang. Let ξ be a set of essential ¯ If N (f¯n ) = 0 then N Pn (f ) = 0. If N (f¯n ) = 0, then n-root representatives for f. N Pn (f ) =

 b∈ξ

N Pn/per(b) ((f per(b) )b )

and

N Φn (f ) =

 N Φn/per(b) ((f per(b) )b ). b∈ξ

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Throughout the paper we will make calculations involving the periodic point theory of (f m )b when b is a representative of a Nielsen class for f¯m . It will be important to see that these calculations are unaffected by certain choices for b. Combining [16, Theorem 2.5, p.80] and Theorem 2.1, we have Proposition 2.12. Let (f, f¯) be a fibre preserving map of a Hurewicz fibration p : E → B. If b, b ∈ Φ(f¯m ) are f¯m Nielsen equivalent, then (f m )b and (f m )b have the same homotopy type, hence, for all m|n, N Pn/m ((f m )b ) = N Pn/m ((f m )b ) and N Φn/m ((f m )b ) = N Φn/m ((f m )b ) All fibre preserving maps (f, f¯) considered in this paper will be such that the base map f¯ is essentially reducible. We plan to extend our results to the situation where the base map is not necessarily essentially reducible elsewhere. 3. The numbers N PnF (f, p) and N ΦF n (f, p) In this section we introduce fibrewise Nielsen type numbers N PnF (f, p) and N ΦF n (f, p) for periodic points ¯ of a fibre preserving map (f, f) : p → p which are lower bounds, respectively, for the minimum number of periodic points of least period n, and of periodic points of all periods dividing n in the fibre homotopy class of f . We also show that N PnF (f, p) and N ΦF n (f, p) can be written as forms of relative Nielsen type numbers for periodic points (Theorem 3.5). Definition 3.1. Let X be a compact connected AN R and let g : X → X be a self map which is an essentially reducible. Then a set of essential irreducible n-representatives of g is a set ξ = {b1 , g(b1 ), ..., g m1 −1 (b1 ), b2 , g(b2 ), ..., g m2 −1 (b2 ), ..., bs , g(bs ), ..., g ms −1 (bs )} ⊆ Φ(g n ) which contains exactly one orbit of periodic points from each essential irreducible orbit class at level m for all m|n, where mi = per(bi ) for each i. Note that #ξ =

 m|n m

m#(IEOm ) =



N Pm (g) = N Φn (g) since any periodic point from essential

m|n

irreducible orbit of f has minimal period m, and that if n = 1 then ξ is simply a set of essential representatives for g as in [11]. Moreover if for all m|n the essential Nielsen classes of g m have unique roots and N Pm (g) = #(IECm (g)), then ξ is a set of essential n-root representatives for g as in [10]. Example 3.2. ([12, 2.3]) Let g : RP3 → RP3 be induced by the map g˜ : S 3 → S 3 given by g˜(ueiφ , veiθ ) = (ue3iφ , ve3iθ ). Then π = π1 (RP3 ) = Z2 , g∗nω = 1π for any choice of x0 and ω so R(g∗nω ) ∼ = Z2 for all n, each algebraic orbit [α]n  consists of the single class {[α]n }, and ιm,n is multiplication by n/m. Now every algebraic periodic point class is essential so g is essentially reducible, and if n = 2r for some r > 0, then

[1]n  is irreducible showing that [1]n  contains at least n periodic points, and [0]n  is reducible. Thus, if n = 2r and ξ is a set of essential irreducible n-representatives of g, then ξ contains one point from [0]1 ,   and m points from [1]m  for each m|n, so #ξ = m|n N Pm (f ) = 1 + m|n m = 2r+1 . But, if n = 2r for some r > 0, then [0]n has more than one roots and N Pn (g) = n = 1 = #(IECn (g)), so g does not satisfy the hypotheses of [10, 3.1], hence g has no set of essential n-root representatives. Let p : E → B be a fibration in which E, B and all fibres F are compact connected AN Rs. Throughout this paper, we will always assume that the base map f¯ : B → B of the fibre preserving map f : E → E is essentially reducible, and that ξ is a set of essential irreducible n-representatives of f¯. Definition 3.3. Let (f, f¯) be a fibre preserving map of p such that f¯ is essentially reducible. Then the fibrewise Nielsen type number of period n of (f, f¯) is

C. Li, Y. Paek / Topology and its Applications 248 (2018) 204–225

N PnF (f, p) =



213

N Pn/per(b) ((f per(b) )b ),

b∈ξ

¯ is and the fibrewise Nielsen type number for the nth iterate of (f, f) N ΦF n (f, p) =



N Φn/per(b) ((f per(b) )b ),

b∈ξ

where ξ is any set of essential irreducible n-representatives for f¯. ¯ = ξ, then f : E → E Proposition 2.12 shows that Definition 3.3 is unaffected by the choice of ξ. Since f(ξ) −1 restricts to a selfmap of p (ξ). We denote this restriction by fξ . Lemma 3.4. Let f : (X, A) → (X, A) be a map of a pair of compact AN R’s with X connected, and n be a given positive integer. If, for all m|n, any essential algebraic orbit of f m is a common essential algebraic orbit of f m and fAm , then N Pn (f ; X, A) = N Pn (fA )

and

N Φn (f ; X, A) = N Φn (fA ).

Proof. Let IEOn (f, fA ) denote the set of common essential irreducible orbits of f n and fAn . By hypothesis and [14, Definition 2.1], all essential irreducible orbits of f n are common essential irreducible orbits of f n and fAn so that IEOn (f ) = IEOn (f, fA ), hence we have from [12, Definition 2.1] and Definition 2.2 that N Pn (f ) = #(IEOn (f )) = #(IEOn (f, fA )) = N Pn (f, fA ), and hence the first equality holds. We also have from [14, Proposition 3.9(i)] that N Φn (f ; X, A) ≥ N Φn (fA ), so, for the second equality, it remains only to show that N Φn (f ; X, A) ≤ N Φn (fA ). Let S ⊆ Orbm|n (fA ) be a set of n-representatives of fA : A → A such that h(S) = N Φn (fA ). We claim that S is also a set of n-representatives of f : (X, A) → (X, A). To see this, let γ ∈ Orbm|n (f )  Orbm|n (fA ) be an essential orbit. If γ ∈ Orbm|n (fA ), then, since S is a set of n-representatives of fA , γ is reducible to some element of S. If γ ∈ Orbm (f ) is an essential orbit with m|n, then, by hypothesis, there is an essential orbit α ∈ Orbm (fA ) so that ν∗ ( α) = γ, and

α is reducible to some element β in S, then γ is reducible to ν∗ ( β) by diagram (2.2). Thus in either case γ is pseudo-reducible to an orbit of S, so S is a set of n-representatives of f : (X, A) → (X, A), we get from [14, Definition 3.5] that N Φn (f ; X, A) ≤ h(S) = N Φn (fA ). 2 The next theorem makes precise the relationship of our work to relative theory [14]. Theorem 3.5. Let (f, f¯) be a fibre preserving map of p such that f¯ is essentially reducible. Then N PnF (f, p) = N Pn (fξ ) = N Pn (f ; E, p−1 (ξ)), and −1 N ΦF (ξ)). n (f, p) = N Φn (fξ ) = N Φn (f ; E, p

Proof. Let ξ = {b1 , f¯(b1 ), ..., f¯m1 −1 (b1 ), b2 , f¯(b2 ), ..., f¯m2 −1 (b2 ), ..., bs , f¯(bs ), ..., f¯ms −1 (bs )} be a set of essen¯ For each orbit b = {b, f¯(b), ..., f¯m−1 (b)} in ξ, the set p−1 ( b) = tial irreducible n-representatives for f. −1 −1 ¯ −1 ¯m−1 (b))} consisting of m(= per(b)) distinct components is an fξ -cycle with {p (b), p (f (b)), ..., p (f length m in p−1 (ξ) and m|n. Therefore, from [13, Definition 3.4] and commutativity of the Nielsen type numbers for periodic points (Theorem 2.1)

214

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N Pn (fξ ) =

s  mi · N Pn/mi ((f mi )bi ) i=1

=

s m i −1  

N Pn/mi ((f mi )f¯j (bi ) )

i=1 j=0

 N Pn/per(b) ((f per(b) )b ) = b∈ξ

= N PnF (f, p). −1 We also have that N ΦF (ξ)) and n (f, p) = N Φn (fξ ) as above. To show that N Pn (fξ ) = N Pn (f ; E, p −1 N Φn (fξ ) = N Φn (f ; E, p (ξ)), we need only show that any essential orbit of f at any level m is a common essential orbit of f m and fξm from Lemma 3.4. Let Am  be an essential orbit of f m , then, from the product theorem for the index (see the remark in [9, 2.7]), A¯m = pε (Am ) is essential, where pε : ε(f m ) → ε(f¯m ) is induced by p. Since f¯ is essentially reducible then A¯m reduces to an essential irreducible class A¯r . Let b be an element of ξ that is contained in A¯r , and k = m/r. Then the application of [9, 2.6] to iterate gives a

exact sequence of essential classes: m/per(b)

ε((f per(b) )b

j∗

pε

) −−−−→ ε(f m ) −−−−→ ε(f¯m ).

Thus there is a Cbk ∈ ε((f r )kb ) with j∗ (Cbk ) = Am . Since r|m, Cbk  is an essential m-periodic orbit of fξ (see [13, p.108 and p.109]) and so Am  = j∗ (Cbk ) is a common essential orbit of f m and fξm (see [14, Definition 2.1]). 2 The fact from [14] that #Pn (f ) ≥ N Pn (f ; E, p−1 (ξ)) and #Φ(f n ) ≥ N Φn (f ; E, p−1 (ξ)) now immediately give us: Proposition 3.6. Let p be a fibration and (f, f¯) a fibre preserving self map such that f¯ : B → B is essentially reducible. Then f has at least N PnF (f, p) periodic points of least period n, and f has at least N ΦF n (f, p) periodic points of period n. Combining Theorem 3.5, [14, 2.9(ii)] and [14, 3.9(ii)], we have Proposition 3.7. Let p be a fibration and (f, f¯) a fibre preserving self map such that f¯ : B → B is essentially reducible. Then N PnF (f, p) ≥ N Pn (f ), and N ΦF n (f, p) ≥ N Φn (f ). Theorem 3.8. (Homotopy invariance) If (f, f¯), (g, g¯) : p → p are fibrewise homotopic and f¯ is essentially F reducible, then N PnF (f, p) = N PnF (g, p), and N ΦF n (f, p) = N Φn (g, p) ¯ = {(ft , f¯t )}0≤t≤1 : (f, f¯)  (g, g¯) be a fibre homotopy, then there exists a homotopy Proof. Let (H, H) m m ¯ ¯ H = {ft }0≤t≤1 : f¯m  g¯m for each positive integer m. Let ξ = {b1 , f¯(b1 ), ..., f¯m1 −1 (b1 ), b2 , f¯(b2 ), ..., f¯m2 −1 (b2 ), ..., bs , f¯(bs ), ..., f¯ms −1 (bs )} ¯ For each i, we choose a point di ∈ Φ(¯ be a set of essential irreducible n-representatives for f. g mi ) that ¯ mi -related to bi . From ordinary Nielsen periodic point theory, it is easy to see that g¯ is essentially is H reducible, and that η = {d1 , g¯(d1 ), ..., g¯m1 −1 (d1 ), d2 , g¯(d2 ), ..., g¯m2 −1 (d2 ), ..., ds , g¯(ds ), ..., g¯ms −1 (ds )}

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215

is a set of essential irreducible n-representatives for g¯. For any path c, let csr (r, s ∈ I) denote the path defined by csr (t) = c((1 − t)r + ts). If b and d are fixed points of f¯m and g¯m respectively and there ¯ m , ω)  ω rel{0, 1}, where (H ¯ m , ω) is the diagonal path exists a path ω from b and to d so that (H m m m ¯ ¯ ¯ of ω in H defined by (H , ω)(t) = H (ω(t), t), then the fibre translation maps τ give a homotopy {τ(H¯ m ,ω)1t ◦ ftm |Fω(t) ◦ τω0t }0≤t≤1 : τ(H¯ m ,ω) ◦ (f m )b  (g m )d ◦ τω as maps from Fb to Fd , furthermore, ¯ m , ω)  ω rel{0, 1} and τω is a homotopy equivalence, (f m )b : Fb → Fb and since τ(H¯ m ,ω)  τω from (H m (g )d : Fd → Fd have the same homotopy type. Thus, for each i, (f mi )bi and (g mi )di have the same homotopy type from the choice of di , and hence N Pn/mi ((f mi )bi ) = N Pn/mi ((g mi )di ) by Theorem 2.1. s  Furthermore, since the proof of Theorem 3.5 gives N PnF (f, p) = mi · N Pn/mi ((f mi )bi ), we get i=1

N PnF (f, p) =

s 

mi · N Pn/mi ((f mi )bi ) =

i=1

s  mi · N Pn/mi ((g mi )di ) = N PnF (g, p). i=1

The second equality is proved in the same way as the first. 2 From the commutativity of ordinary Nielsen type numbers for periodic points, we have that Proposition 3.9. (Commutativity) Suppose that (f, f¯) : p → q and (g, g¯) : q → p are fibre preserving maps between two fibrations p and q such that g¯f¯ is essentially reducible (and hence so is f¯g¯). Then F N PnF (gf, p) = N PnF (f g, q), and N ΦF n (gf, p) = N Φn (f g, q). The following proposition is immediate from Theorem 3.8 and Proposition 3.9. Proposition 3.10. (Homotopy type invariance) Suppose that (j, ¯j) : p → q is a fibrewise homotopy equivalence and that (f, f¯) : p → p and (g, g¯) : q → q have the same fibrewise homotopy type (i.e., gj, jf : p → q are fibrewise homotopic). If f¯ is essentially reducible (and hence so is g¯), then N PnF (f, p) = N PnF (g, q), and F N ΦF n (f, p) = N Φn (g, q). 4. Relations between the fibrewise Nielsen type numbers In this section we will establish some relations between the two fibrewise Nielsen type numbers for periodic points studied in this paper and the Nielsen type number of the nth iterate of f , i.e., between the numbers N PnF (f, p) and N ΦF (f, p) and the number NF (f n , p). We also consider conditions so that N ΦF n (f, p) can n be found as sum b∈ξ N ((f n )b ) of ordinary Nielsen numbers on fibres, using the results from [9]. The first result is obvious. F Proposition 4.1. N ΦF 1 (f, p) = N P1 (f, p) = NF (f, p).

Theorem 4.2. If (f, f¯) : p → p is a fibre preserving map such that f¯ is essentially reducible, then N ΦF n (f, p) ≥



F N Pm (f, p).

m|n

If, furthermore, all fibres F are essentially reducible, then N ΦF n (f, p) =

 m|n

F N Pm (f, p).

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216

Proof. Let ξ be a set of essential irreducible n-representatives for f¯. We have that N ΦF n (f, p) = N Φn (fξ )  ≥ N Pm (fξ ) m|n



=≥

F N Pm (f, p),

m|n

where both equalities follow from Theorem 3.5, and the inequality follows from [13, Proposition 3.11(iii)]. Furthermore, if all fibres are essentially reducible, the equality needed holds by [13, Proposition 4.6]. 2 If we apply the Möbius inversion formula to the second part of Theorem 4.2, then we obtain the following corollary. ¯ : Corollary 4.3. Let p : E → B be a fibration in which all fibres F are essentially reducible, and let (f, f) ¯ p → p be the fibre preserving map such that f is essentially reducible. Let n be a positive integer, and P(n) = {p1 , ..., pk } the set of all primes dividing n, then N PnF (f, p) =



(−1)#τ N ΦF n:τ (f, p),

τ ⊆P(n)

where n : τ = n

 p∈τ

p−1 .

The following results compare the fibrewise Nielsen type number N ΦF n (f, p) for periodic points with the n n fibrewise Nielsen type number NF (f , p) of f . We also give conditions under which the number N ΦF n (f, p) equals both NF (f n , p) and N Φn (f ). Proposition 4.4. If (f, f¯) : p → p is a fibre preserving map such that f¯ is essentially reducible, then n N ΦF n (f, p) ≥ NF (f , p). Proof. Let ξ be a set of essential irreducible n-representatives for f¯. We claim that ξ contains a set of essential representatives for f¯n . Let ε(f¯n ) = {A¯n1 , A¯n2 , ..., A¯nN (f¯n ) }. Since f¯ is essentially reducible, for each ¯m i, A¯ni contains an essential irreducible class A¯m i with m|n, and by Definition 3.1 ξ contains a point bi in Ai .  n Then ξ = {b1 , b2 , ..., b ¯ } is a set of essential representatives for f¯ which is contained in ξ. Then N (f )

N ΦF n (f, p) =

 N Φn/per(b) ((f per(b) )b ) b∈ξ

≥



N Φn/per(b) ((f per(b) )b )

b∈ξ 

≥



N ((f n )b ) = NF (f n , p),

b∈ξ 

where the second inequality follows from the inequality N Φm (g) ≥ N (g m ), and the last equality follows from [11, Definition 6.1]. 2   Example 4.5. Let S 3 × R = {(x, y)} = { (ueiφ , veiθ ), y }. Consider the equivalence relation on S 3 × R and the resulting quotient space S determined by the relations (x, y) ∼ (−x, −y), and (x, y) ∼ (x, y + l) for

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l ∈ Z. Now the obvious fibration R → S 3 × R → S 3 induces a Hurewicz fibration S 1 → S → RP3 . The     correspondence (ueiφ , veiθ ), y → (ue3iφ , ve3iθ ), y induces a well defined fibre preserving map f on S, and the base map f¯ = g : RP3 → RP3 is as in Example 3.2. Let n = 2r , and ξ be a set of essential irreducible n-representatives for g. As shown in Example 3.2, ξ consists of one point from [0]1 , and m points from [1]m  for each m|n. We have that for all m, the degree of (f m )b is 1 if b is a point from

[0]m  and the degree of (f m )b is −1 if b is a point from [1]m . For a map h : S 1 → S 1 of degree d, we abbreviate the numbers N (h), N Pk (h) and N Φk (h) by N (deg d), N Pk (deg d) and N Φk (deg d) respectively. Since N Pk (deg − 1) = 0 for all k > 1 and N Φk (deg − 1) = 2 for all k > 0, for n = 2r , we have N PnF (f, p) =



N Pn/per(b) ((f per(b) )b )

b∈ξ



= N Pn (deg + 1) +

m · N Pn/m (deg − 1)

m|n

= n · N P1 (deg − 1) = 2r · 2 = 2r+1 , and N ΦF n (f, p) =



N Φn/per(b) ((f per(b) )b )

b∈ξ

= N Φn (deg + 1) + =





m · N Φn/m (deg − 1)

m|n

2m = 2r+2 − 2.

m|n

But, for all n, N (f¯n ) = N (g n ) = 2 and a set of essential representatives for f¯n consists of one point b0 in [0]n and one point b1 in [1]n , so we have from [11, 6.1] that NF (f n , p) = N ((f n )b0 ) + N ((f n )b1 ) = N (deg + 1) + N (deg − 1) = 2. Note that (f, f¯) does not satisfy the hypotheses of [10, 3.4], so one cannot use formulae from [10] to calculate N Pn (f ) and N Φn (f ). Next lemma gives conditions so that N ΦF n (f, p) can be found as a sum of ordinary Nielsen numbers N ((f n )b ) on fibres. Lemma 4.6. Suppose that p : E → B is a fibration in which all fibres F have essential torality, and are ¯ : p → p is a fibre preserving map such that f¯ essentially reducible to the GCD. Suppose further that (f, f) ¯ If for each b ∈ ξ, (f n )b is essentially reducible. Let ξ be a set of essential irreducible n-representatives for f.  is weakly Jiang and N ((f n )b ) = 0, then N ΦF N ((f n )b ). n (f, p) = b∈ξ n/per(b)

Proof. By [9, 5.1], N Φn/per(b) ((f per(b) )b ) = N ((f per(b) )b follows from Definition 3.3. 2

) = N ((f n )b ) for each b ∈ ξ, then the result

Here is an example in which Lemma 4.6 is used to calculate the fibrewise Nielsen type number N ΦF n (f, p) for the nth iterate.

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Example 4.7. Consider a solvmanifold in [10, Example 4.3]. Let R3 = R2 × R = {(x, y)}. Consider the equivalence relation on R3 and the resulting quotient space S determined by the relations (x, y) ∼ (x + k, A(k)y) for k ∈ Z2 , and (x, y) ∼ (x, y + l) for l ∈ Z, where homomorphism A : Z2 → Z2 ∼ = {1, −1} is a+b 1 defined by the rule A((a, b)) = (−1) . Then S is the total space of a Hurewicz fibration S → S → T 2 induced from the obvious fibration R → R2 × R → R2 . As we shall see also in Example 4.9, note that this fibration satisfies the naïve addition conditions, and since tori are nilmanifolds the base and all fibres are essentially toral, are essentially reducible to the GCD, and are weakly Jiang (see [9, 4.12, 4.16, and 4.21]). We construct next just such a fibre preserving map, but it is not same as the fibre preserving map considered in [10, Example 4.3]. Consider the following matrices     0 1 −1 0 2 Y = and Y = . −1 0 0 −1 Note that Y 4 = I, and that A(Y (k)) = A(k) for any k ∈ Z2 and that the correspondence (x, y) → (Y (x), 2y) induces a well defined fibre preserving map f on S (note that the correspondence (x, y) → (Y (x), −y) is used in [10, 4.3] instead of (x, y) → (Y (x), 2y)). Since f n ((x, y)) = (Y n (x), 2n y), then f¯ has period 4, and so N Pk (f¯) = 0 for k > 4. From the formula N (f¯n ) = |det(Y n − I)| (see [4]), we have that N (f¯) = N (f¯3 ) = 2, N (f¯2 ) = 4, and N (f¯4 ) = 0. From the fact that T 2 is a Jiang space and has uniqueness of roots for essential classes, and N (f¯3 ) = 0 it follows that both Nielsen classes of f¯3 reduce to level 1, thus N P3 (f¯) = 0. Now we show that p and (f, f¯) are satisfy the hypotheses of Lemma 4.6 to calculate N ΦF n (f, p). To calculate N ΦF (f, p), we need more specific information about how the degrees of f and (f 2 )b vary over b n 2 2 2 ¯ ¯  ¯ ¯  points b ∈ Φ(f ) or Φ(f ). Let b ∈ R represent a point b = [b] ∈ Φ(f ) or Φ(f ). For example, when b ∈ Φ(f¯) then Y (b) = b + k for some k ∈ Z2 . So f ((b, y)) = (Y (b), 2y) = (b + k, 2y) ∼ (b, A(−k)(2y)) = (b, 2A(k)y) so that the degree of fb is 2A(k). Similarly, for b ∈ Φ(f¯2 ) we get that f 2 sends (b, y) to (b, 4A(l)y) where l ∈ Z2 is such that Y 2 (b) = b + l, and thus the degree of (f 2 )b is 4A(l). ¯ By the consideration of matrices Y and Now we find a set of essential irreducible n-representatives for f. 1 1 1 2 2 ¯ Y we see that Φ(f ) = {b1 = [(0, 0)], b2 = [( 2 , 2 )], b3 = [(0, 2 )], b4 = [( 12 , 0)]}, with b1 and b2 in Φ(f¯), and b3 and b4 being of minimal period 2. Note also that since N Φ2 (f¯) = N (f¯2 ) = 4, each point of Φ(f¯2 ) belongs to its own Nielsen class, we have that ξ = Φ(f¯2 ) is a set of essential irreducible n-representatives for any even n, and ξ1 = {b1 , b2 } is a set of essential irreducible n-representatives for any odd n. Now Y (b1 ) = b1 , and Y (b2 ) = b2 + (0, −1), so deg(fb1 ) = 2 and deg(fb2 ) = −2 by the above analysis. Thus, if n is odd, then since N ((f n )b ) = N (deg+2n ) = 0 for each b ∈ ξ1 , p and (f, f¯) are satisfy the hypothesis of Lemma 4.6, so we have that n n n n n+1 N ΦF . n (f, p) = N ((f )b1 ) + N ((f )b2 ) = (2 − 1) + (2 + 1) = 2

Now Y 2 (b1 ) = b1 , Y 2 (b2 ) = b2 + (−1, −1), Y 2 (b3 ) = b3 + (0, −1), and Y 2 (b4 ) = b4 + (−1, 0), so for each i, deg((f 2 )bi ) = +4. Thus, if n is even and not a multiple of 4, then N ((f n )b ) = N (deg+4n ) = 0 for each b ∈ ξ, thus we also have from Lemma 4.6 that N ΦF n (f, p)

4  = N ((f n )bi ) = (2n − 1) + (2n − 1) + (2n + 1) + (2n + 1) = 2n+2 ; i=1

if n is a multiple of 4, then as above we have that N ΦF n (f, p) =

4  N ((f n )bi ) = 4N (deg + 2n ) = 4(2n − 1) = 2n+2 − 4. i=1

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Proposition 4.8. Suppose that p : E → B is a fibration satisfying the naïve addition conditions in which B ¯ : p → p be a fibre and all fibres F are essentially toral, and are essentially reducible to the GCD. Let (f, f) ¯ If, furthermore, f and f¯ satisfy preserving map, and ξ a set of essential irreducible n-representatives for f. n n n n ¯ ¯ ¯ ¯ (i) f is weakly Jiang, and N (f ) = 0 (i.e., N (f ) = R(f )), (ii) for each b ∈ ξ, (f n )b is weakly Jiang, and N ((f n )b ) = 0. Then for all m|n, m m N ΦF m (f, p) = N Φm (f ) = NF (f , p) = N (f ) =



N ((f m )b ),

b∈ξm

where ξm = ξ



Pm (f¯).

Proof. It is not hard to see that for each m|n, ξm is a set of essential irreducible m-representatives for f¯. The conditions (a) that B and all fibres F are essentially reducible and (b) N (f¯n ) = R(f¯n ) and for each b ∈ ξ, N ((f n )b ) = R((f n )b ) together imply that N (f¯m ) = R(f¯m ) and for each b ∈ ξm , N ((f m )b ) = R((f m )b ). Thus we need only prove the proposition for m = n. As in the proof of [9, 5.1], we have that    #ξ = m|n N Pm (f¯) = m|n #(IECm (f¯)), and that a function φ : m|n IECm (f¯) → ε(f¯n ), defined by φ(C d ) = γd,n (C d ), is a bijection. This implies that ξ is also a set of essential representatives for f¯n , and since ¯ are satisfy the hypothesis of Lemma 4.6, we have that p satisfies the naïve addition conditions and (f, f)  n from [11, 4.4] and Lemma 4.6 that N (f ) = N ((f n )b ) = N ΦF n (f, p). Combining this with the inequalities b∈ξ n F n n N ΦF n (f, p) ≥ N Φn (f ) ≥ N (f ) N Φn (f, p) ≥ NF (f , p) ≥ N (f ) which follow from Proposition 3.7, Proposition 4.4, [15, 3.12] and [11, 7.3], completes our proof. 2

Example 4.9. Let (f, f¯) be the fibre preserving map of the fibration S 1 → S → T 2 defined in Example 4.7. If n is a multiple of 4, then N (f¯n ) = 0 so condition (i) of Proposition 4.8 is not satisfied for f¯. We will show n n that if n is not a multiple of 4, then N ΦF n (f, p) = N Φn (f ) = NF (f , p) = N (f ); but if n is a multiple of n n 4, then N ΦF n (f, p) > N Φn (f ) > NF (f , p) = N (f ) = 0. By the consideration of Example 4.7 we see that n ¯ if n is not a multiple of 4, then N (f ) = 0, and p and (f, f¯) are satisfy the hypotheses of Proposition 4.8. Thus we get that

n n N ΦF n (f, p) = N Φn (f ) = NF (f , p) = N (f ) =

 2n+1 2

n+2

if n is odd, if n ≡ 2 (mod 4).

But if n is a multiple of 4, then N (f¯n ) = 0 so we have that NF (f n , p) = N (f n ) = 0 from [11, Definition 6.1] and the product theorem for the index. Now we use [9, Theorem 5.8] to calculate the Nielsen type number N Φn (f ) for 4|n. When n is a multiple of 4, then n is of the form n = 2r m with m odd and r > 1. In this case, M (f, n) = {2m}, and since for every b ∈ ξ = Φε (f¯2m ), N ((f 2m )b ) = 0, then f 2m is weakly Jiang from [9, Corollary 4.26], and we have from [9, Theorem 5.8] and Example 4.7 that N Φn (f ) = N (f 2m ) = 22m+2 < 2n+2 − 4 = N ΦF n (f, p). F Now we calculate N P12 (f, p) in two different ways. 2+2 4+2 = 16, N ΦF −4 = First we use the formula from this example to calculate N ΦF 2 (f, p) = 2 4 (f, p) = 2 F F 8 14 60, N Φ6 (f, p) = 2 = 256, and N Φ12 (f, p) = 2 − 4 = 16380, and we get from Corollary 4.3 that F F F F N P12 (f, p) = N ΦF 12 (f, p) − N Φ6 (f, p) − N Φ4 (f, p) + N Φ2 (f, p)

= 16380 − 256 − 60 + 16 = 16080.

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Next we have from [9, Theorem 5.1] that N P12 (fb1 ) = N P12 (deg + 2) = (212 − 1) − (26 − 1) − (24 − 1) + (22 − 1) = 4020, N P12 (fb2 ) = N P12 (deg − 2) = 4020, N P6 ((f 2 )b3 ) = N P6 ((f 2 )b4 ) = N P6 (deg − 4) = 4020, and from Definition 3.3 we calculate F N P12 (f, p) = N P12 (fb1 ) + N P12 (fb2 ) + N P6 ((f 2 )b3 ) + N P6 ((f 2 )b4 )

= 4020 · 4 = 16080, and hence the same value as first. But N (f 12 ) = 0 and M (f, 12) = {6}, so we get that N P12 (f ) = 0 and N Φ12 (f ) = 26+2 = 256. 5. Product formulae for N PnF (f, p) and N ΦF n (f, p) Without loss to the homotopy type of f¯ we can assume that f¯ is basepoint preserving with respect to a distinguished base point 0 ∈ Φ(f¯). Let f0 be the restriction of f to p−1 (0) the map on the principal fibre as in [10]. In the presence of Nielsen type number fibre uniformity many of the above considerations are F substantially simplified. In particular we obtain conditions so that N ΦF n (f, p) can be found as N Φn (f, p) = ¯ N Φn (f )N Φn (f0 ), the product of the Nielsen type numbers for periodic points on the base and the principal fibre. Proposition 5.1. Let (f, f¯) be a fibre preserving map of p such that f¯ is essentially reducible, and f has Nielsen type number fibre uniformity (i.e., for a fixed n and all m|n, N Pn/m ((f m )b ) and N Φn/m ((f m )b ) are independent of b ∈ Φ(f¯m ), then N PnF (f, p) =



N Pm (f¯)N Pn/m (f0m ),

m|n

 N ΦF N Pm (f¯)N Φn/m (f0m ). n (f, p) = m|n

Proof. By fibre uniformity, N Pn/per(b) ((f per(b) )b ) = N Pn/per(b) ((f per(b) )0 ), N Φn/per(b) ((f per(b) )b ) = N Φn/per(b) ((f per(b) )0 ) for each b ∈ ξ. By the definition of ξ there are N Pm (f¯) of these terms with per(b) = m for each m|n. The result follows from Definition 3.3. 2 We state the results of the first example as a proposition to show how to use the Proposition 5.1. Proposition 5.2. Let P

T 2 → T 4 = T 2 × T 2 −−−−→ T 2 and f = f0 × f¯ : T 4 → T 4 be as in [10, Example 4.10]. Then

C. Li, Y. Paek / Topology and its Applications 248 (2018) 204–225

N P1F (f, p) = 2,

N P2F (f, p) = 10,

221

N P3F (f, p) = N P6F (f, p) = 6,

and N PnF (f, p) = 0 for n = 4, 5 and n > 6. N ΦF n (f, p) are periodic with period six, and ⎧ ⎪ 2 ⎪ ⎪ ⎪ ⎨12 N ΦF n (f, p) = ⎪8 ⎪ ⎪ ⎪ ⎩ 24

if n ≡ 1 (mod 6) or 5 (mod 6), if n ≡ 2 (mod 6) or 4 (mod 6), if n ≡ 3 (mod 6), if 6|n.

Proof. We first calculate the numbers N PnF (f, p). Since f has Nielsen type number fibre uniformity, we get  from Proposition 5.1 that for all n, N PnF (f, p) = N Pm (f¯)N Pn/m (f0m ). It is easy to see that N P1 (f¯) = 1, m|n

N P2 (f¯) = 2, N P3 (f¯) = 3, and N P4 (f¯) = N P5 (f¯) = 0, and, since f¯ has period six, N Pm (f¯) = 0 for  all m ≥ 6. Therefore the above sum reduces to N Pm (f¯)N Pn/m (f0m ), so it remains only to calculate m≤3

N Pn/m (f0m ) for m ≤ 3. Now f0 and f03 have period four, and f02 has period 2. Then it is not hard to get that N Pk (f0m ) = 0 for m = 1, 2, 3 if k > 2. Therefore, if n > 6, then N Pn/m (f0m ) = 0 for every m ≤ 3, thus  we get that N PnF (f, p) = N Pm (f¯)N Pn/m (f0m ) = 0. For n ≤ 6, we get from the above analysis m≤3

N P1F (f, p) = N P1 (f¯)N P1 (f0 ) = 1 · 2 = 2, N P2F (f, p) = N P1 (f¯)N P2 (f0 ) + N P2 (f¯)N P1 (f02 ) = 1 · 2 + 2 · 4 = 10, N P3F (f, p) = N P1 (f¯)N P3 (f0 ) + N P3 (f¯)N P1 (f03 ) = 1 · 0 + 3 · 2 = 6, N P4F (f, p) = N P1 (f¯)N P4 (f0 ) + N P2 (f¯)N P2 (f02 ) = 1 · 0 + 2 · 0 = 0, N P5F (f, p) = N P1 (f¯)N P5 (f0 ) = 1 · 0 = 0 and finally N P6F (f, p) = N P1 (f¯)N P6 (f0 ) + N P2 (f¯)N P3 (f02 ) + N P3 (f¯)N P2 (f03 ) = 1 · 0 + 2 · 0 + 3 · 2 = 6. Next we can use the formula N ΦF n (f, p) =



F N Pm (f, p) from Theorem 4.2 to calculate the numbers

m|n F N ΦF n (f, p). Since all N Pm (f, p) = 0 with m  6 contribute nothing, we get

N ΦF n (f, p) =

 m|6,n



F N Pm (f, p) =

F N Pm (f, p),

m|(6,n)

where (6, n) is the greatest common divisor of 6 and n. Since ⎧ ⎪ 1 if n ≡ 1 (mod 6) or 5 (mod 6), ⎪ ⎪ ⎪ ⎨2 if n ≡ 2 (mod 6) or 4 (mod 6), (6, n) = ⎪3 if n ≡ 3 (mod 6), ⎪ ⎪ ⎪ ⎩ 6 if 6|n, we calculate F N ΦF n (f, p) = N P1 (f, p) = 2 if n ≡ 1 (mod 6) or 5 (mod 6),

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222

N ΦF n (f, p) =



F N Pm (f, p) =

m|2

N ΦF n (f, p) =



F N Pm (f, p) =

m|3

N ΦF n (f, p) =



F N Pm (f, p) =

m|6

 m=1,2



m=1,3

F N Pm (f, p) = 2 + 10 = 12 if n ≡ 2 (mod 6) or 4 (mod 6), F N Pm (f, p) = 2 + 6 = 8 if n ≡ 3 (mod 6), and



m=1,2,3,6

F N Pm (f, p) = 2 + 10 + 6 + 6 = 24 if 6|n.

2

To compare N PnF (f, p) and N ΦF n (f, p) with N Pn (f ) and N Φn (f ) for the first example, we get from the formula N (f n ) = |det(An − I)| and [9, 5.8] that N P1 (f ) = N (f ) = 2, N P2 (f ) = N (f 2 ) − N (f ) = 10, N P3 (f ) = N (f 3 ) − N (f ) = 6, and, for all n > 3, N Pn (f ) = 0. Thus N P6F (f, p) = 6 > 0 = N P6 (f ). If n is not a multiple of 6 then N (f¯n ) = 0 so we get from [10, 3.6] and Definition 3.3 that N ΦF n (f, p) = N Φn (f ). But if n is a multiple of 6 then we get from [9, 4.6] and the collection of the numbers N Pm (f ) that   N Φn (f ) = m|n N Pm (f ) = m≤3 N Pm (f ) = 2 + 10 + 6 = 18 < 24 = N ΦF n (f, p). Corollary 5.3. Let (f, f¯) be a fibre preserving map such that f¯ is essentially reducible, and f has Nielsen type number fibre uniformity as in Proposition 5.1. Suppose further that f0 is essentially toral, essentially reducible to the GCD, that f0n is weakly Jiang and N (f0n ) = 0. Then n ¯ ¯ N ΦF n (f, p) = N Φn (f )N Φn (f0 ) = N Φn (f )N (f0 ).

Proof. By the first hypothesis, the second equality in Proposition 5.1 holds for N ΦF n (f, p). Furthermore, m n since for all m|n, N Φn/m (f0 ) = N Φn (f0 ) = N (f0 ) by [10, Lemma 4.8] and [9, Theorem 5.1], we have N ΦF n (f, p)

=



 ¯ N Pm (f ) N Φn (f0 ) = N Φn (f¯)N Φn (f0 ) = N Φn (f¯)N (f0n ).

2

m|n

We give another fibre map f = f0 × f¯ : T 4 → T 4 for which f0 and f¯ are not periodic. Example 5.4. We reconsider the trivial fibration P

T 2 → T 4 = T 2 × T 2 −−−−→ T 2 where p is projection on the second factor as in [10, Example 4.10]. Let f = f0 × f¯ : T 4 → T 4 which is induced by the matrix ⎡

2 ⎢0 A=⎣ 0 0

⎤ 0 0 0 0 ⎥ . 2 0 ⎦ 1 −1

0 3 0 0

So the map f0 on the principal fibre is induced by the 2 × 2 matrix B in the upper left, and the induced map f¯ on the base from the 2 × 2 matrix C in the lower right. For all n, since N (f0n ) = |det(B n − I)| = n ¯ (2n − 1)(3n − 1) = 0, we have from Corollary 5.3 that N ΦF n (f, p) = N Φn (f )N (f0 ). We will show that if n F F is odd, then N Φn (f, p) = N Φn (f ); but if n is even, then N Φn (f, p) > N Φn (f ). We have that  n

C =

2n

2 −(−1) 3 n

n

 0 , (−1)n

and hence N (f¯n ) = |det(C n − I)| =

 2n+1 − 2 if n is odd, 0

if n is even.

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If n is odd, then N (f¯n ) = 0, so we have from Proposition 4.8 and [9, Theorem 2.8] that n n N ΦF n (f, p) = N Φn (f ) = NF (f , p) = N (f )

= N (f¯n )N (f0n ) = 2(2n − 1)2 (3n − 1). But if n is even, then n is of the form n = m · 2r with m odd and r > 0, and, since N (f¯n ) = N (f n ) = 0 and M (f¯, n) = M (f, n) = {m}, we have from Corollary 5.3 and [9, Theorem 5.8] that n n m n n ¯ ¯m N ΦF n (f, p) = N Φn (f )N (f0 ) = N (f )N (f0 ) = 2(2 − 1)(2 − 1)(3 − 1),

N Φn (f ) = N (f m ) = N (f¯m )N (f0m ) = 2(2m − 1)2 (3m − 1) < N ΦF n (f, p), and that NF (f n , p) = N (f n ) = 0, N Pn (f ) = 0. As a special case of this, for n = 12, we have from the above formulae that N Φ12 (f ) = 2(23 − 1)2 (33 − 1) = 2548, N P12 (f ) = 0; 1 2 2 N ΦF 2 (f, p) = 2(2 − 1)(2 − 1)(3 − 1) = 48, 1 4 4 N ΦF 4 (f, p) = 2(2 − 1)(2 − 1)(3 − 1) = 2400, 3 6 6 N ΦF 6 (f, p) = 2(2 − 1)(2 − 1)(3 − 1) = 662480

and finally 3 12 N ΦF − 1)(312 − 1) = 30467455200 12 (f, p) = 2(2 − 1)(2

giving from Corollary 4.3 that F F F F N P12 (f, p) = N ΦF 12 (f, p) − N Φ6 (f, p) − N Φ4 (f, p) + N Φ2 (f, p) = 30466790368.

Our final result gives conditions under which the product formula n ¯ N ΦF n (f, p) = N Φn (f )N Φn (f0 ) holds without the assumption that N (f0 ) = 0. From the equality #ξ =  N Pm (f¯) = N Φn (f¯) and Definition 3.3, it is easy to see that m|n

Proposition 5.5. Suppose that f, f¯ and p are as in Proposition 5.1. Suppose further that N Φn/m (f0m ) are independent of m|n with N Pm (f¯) = 0. Then N ΦF n (f, p) =



N Pm (f¯)N Φn (f0 ) = N Φn (f¯)N Φn (f0 ).

m|n



N Pm (f¯)N Φn/m (f0m ). Further    F m ¯ ¯ N Pm (f )N Φn/m (f0 ) = N Pm (f ) N Φn (f0 ) = more, by the second hypothesis, we have N Φn (f, p) = Proof. The first hypothesis and Proposition 5.1 give that N ΦF n (f, p) =

m|n

m|n

N Φn (f¯)N Φn (f0 ).

m|n

2

Example 5.6. Let P

T 2 → T 4 = T 2 × T 2 −−−−→ T 2

224

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and f = f0 × f¯ : T 4 → T 4 be as in [10, Example 4.10]. As shown in the proof of Proposition 5.2, N Pm (f¯) = 0 for all m > 3. Thus we get from Proposition 5.1 that N ΦF n (f, p) =



N Pm (f¯)N Φn/m (f0m ) =



N Pm (f¯)N Φn/m (f0m ).

m≤3

m|n

We will show that (f, f¯) satisfies the hypotheses of Proposition 5.5 for all n. From the above sum we need only show that for a fixed n, N Φn/m (f0m ) is independent of m|n with m ≤ 3. We calculate that if n is odd, then N Φn/m (f0m ) = N Φn (f0 ) = N (f0n ) = 2 for each m|n from [10, Lemma 4.8], and that if n is even, then N Φn/m (f0m ) = N Φn (f0 ) = N (f02 ) = 4 for m = 1 and 2 from [9, Theorem 5.8] as needed. Thus the product ¯ ¯n ¯ formula N ΦF n (f, p) = N Φn (f )N Φn (f0 ) holds for all n. Since N (f ) = 0 if n is not a multiple of 6, and f has period six, then we have from [9, Theorem 5.1] that ⎧ ⎪ ⎪ ⎨1 if n ≡ 1 (mod 6) or 5 (mod 6), n n N Φn (f¯) = N (f¯ ) = |det(C − I)| = 3 if n ≡ 2 (mod 6) or 4 (mod 6), ⎪ ⎪ ⎩4 if n ≡ 3 (mod 6).  If n is a multiple of 6, then N Φn (f¯) = N Pm (f¯) = m|n



N Pm (f¯) = 6. Therefore we calculate

m=1,2,3

⎧ ⎪ 1·2=2 if n ≡ 1 (mod 6) or 5 (mod 6), ⎪ ⎪ ⎪ ⎨3 · 4 = 12 if n ≡ 2 (mod 6) or 4 (mod 6), ¯ N ΦF n (f, p) = N Φn (f )N Φn (f0 ) = ⎪ 4·2=8 if n ≡ 3 (mod 6), ⎪ ⎪ ⎪ ⎩ 6 · 4 = 24 if 6|n, confirming the result in Proposition 5.2. It is natural to ask if the equalities in Theorem 3.5 and hence the inequalities N Pn (f ) ≤ N PnF (f, p) and ¯ N Φn (f ) ≤ N ΦF n (f, p) in Proposition 3.7 hold without the assumption that f is essentially reducible. The answer is a resounding now as the following example shows. This also illustrates why the hypothesis that f¯ is essentially reducible is needed in Proposition 3.7 (and hence also Theorem 3.5). Example 5.7. Consider the self map on X = S 2 ∨ S 3 as defined in [15, 3.1]. Let g : S 3 → S 3 be a map of degree 2, x0 be a fixed point of g, and X = S 2 ∨ S 3 , with S 2 ∩ S 3 = {x0 }; let f¯ : X → X be a map whose restriction to S 2 × {x0 } is the identity map and whose restriction to {x0 } × S 3 is g (note that the symbol f is used in [15, 3.1] instead of f¯). Consider the trivial fibration P

S 1 → S 1 × X −−−−→ X, where p is projection on X. Let f = f0 × f¯ : S 1 × X → S 1 × X where the map f0 : S 1 → S 1 on the principal fibre has degree d = +1. The Lefschetz number L(f¯m ) of f¯m is 2 − 2m so L(f¯) = 0 and, for every m > 1, L(f¯m ) = 0; now since π1 (X) and hence Coker(1 − f¯mω ) = 0 for all m, N (f¯) = 0 and, for every m > 1, N (f¯m ) = 1 (see [15, 3.1]). Thus, for every m > 1, the unique algebraic orbit [0]m  at level m is essential  but reduces to the inessential algebraic orbit [0]1 . Hence, for every n > 1, m|n N Pm (f¯) = 0 and any set of essential irreducible n-representatives for f¯ is empty, so N PnF (f, p) = 0 and N ΦF n (f, p) = 0, while N Φn (f ) ≥ N (f n ) = N (f¯n )N (f0n ) = N (f0n ) = |L(f0n )| = |1 − dn | > 0, therefore N Φn (f ) > N ΦF n (f, p) = 0.

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Acknowledgements This work was motivated by the results and observations in [11,10]. The definitions of the two fibrewise periodic Nielsen type numbers here are inspired by the work in [10]. The authors would like to thank P.R. Heath, E.C. Keppelmann and P. Wong for their suggestion about the main point of this paper. Finally, the authors would like to thank the referee for many valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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