Product formula for the fixed point index of a fibre preserving map defined locally

Topology and its Applications 267 (2019) 106895

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

Product formula for the fixed point index of a fibre preserving map defined locally Changbok Li 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 10 April 2019 Received in revised form 6 September 2019 Accepted 17 September 2019 Available online 23 September 2019 MSC: 55M20 55R05

a b s t r a c t There is well known product formula for the index of a fixed point class of a fibre preserving map which shows that the index in the total space is the product of indices in the fibre and base. In this paper we generalize these ideas to the setting of fibre preserving maps defined locally. Let (E, p, B) be a fibre space with E, B and all fibres compact connected ANR’s, V be an open subset in B, and U = p−1 (V ). Let f : U → E be a fibre preserving map, i.e., a map for which there exists a map f¯ : V → B with pf = f¯pU,V , where pU,V : U → V is the restriction of p. We study the structure of the fixed point classes of f , and prove product formula for the index of a fixed point class of f . © 2019 Elsevier B.V. All rights reserved.

Keywords: Nielsen numbers Fibre spaces Admissible maps Maps defined locally

1. Introduction Nielsen fixed point theory uses the notion of fixed point classes to give a lower bound for the cardinality of the fixed point set of a self map. In this paper we are concerned with using fibre space theory to assist in the calculation of the Nielsen number of a locally defined map. Since the introduction of the so called naïve product formula for the Nielsen number of a fibre preserving map [4], there have been a number of attempts to use fibre spaces as a means of computing Nielsen numbers (see for example [2,5,10–18, etc.]). The idea has been to write the Nielsen number of the self map of the total space in terms of appropriate Nielsen numbers of the maps between the base and fibre. Let p : E → B be a fibration of compact connected 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 [13] Heath, Keppelmann and Wong introduced the naïve addition formula  N (f ) = N (fb ), expressing the Nielsen number of the fibre preserving map as a simple sum of Nielsen b∈ξ

numbers on the fibres, where ξ is a set which contains exactly one point from each essential Nielsen class E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2019.106895 0166-8641/© 2019 Elsevier B.V. All rights reserved.

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of f¯, and fb denotes the restriction f |p−1 (b) : p−1 (b) → p−1 (b) of f to the fibre. The fundamental theorem, which underlies all product formulae and addition formulae for Nielsen numbers of fibre preserving maps, is the product theorem for the index which gives that a class is essential in the total space iff it is essential in both base and fibre (see Theorem 2.7). Meanwhile, Fadell and Husseini [7] developed a local Nielsen fixed point theory for compactly fixed maps defined on open subsets of ENRs (Euclidean Neighbourhood Retracts) using the index theory developed by [6]. Andres and Wong [1] gave a generalization of the local Nielsen theory of [7] by using an appropriate index theory developed by Granas [8,9] for ANRs. The main objective of this paper is to extend product theorem for the index of a fixed point class of fibre preserving maps to a local setting. Its motivation is the fact that it is needed as indispensable material for an extension of the results from [13], to fibre preserving maps defined locally. We plan to introduce the naïve addition formula, expressing the Nielsen number of the fibre preserving map defined locally as a simple sum of Nielsen numbers on the fibres. 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 describe Reidemeister classes of a locally defined map. In order to describe empty fixed point class, we will use a modified fundamental group approach unlike [1] which used the covering space approach. In Section 4 we discuss the properties of fixed point classes of a fibre preserving map defined locally. Finally we prove, in Section 5, the product formula for the index of a fixed point class of a fibre preserving map defined locally. 2. Review In this section we briefly review some of the concepts from Nielsen fixed point theory for fibrations, and the local Nielsen fixed point theory that are needed in the study of the index of Nielsen fixed point class of a fibre preserving map defined locally. 2.1. Product formula for the index of a fixed point class of fibre preserving maps We use a modified fundamental group approach as in [18]. Throughout the paper unless otherwise stated all spaces X, Y , etc. will be compact connected ANR’s. Suppose that f : X → X is a self map of X. Let Φ(f ) = {x ∈ X : f (x) = x} denote the fixed point set of f . If a path c is homotopic to a path d relative to the end points, then we will write c  d. We say that x, y ∈ Φ(f ) are f Nielsen equivalent, denoted x ∼f y, provided that there is a path c from x to y so that c  f (c). We will denote the set of equivalence classes thus generated by Φ (f ), and call them the set of all geometric Nielsen classes for f . Standard index theory (see for example [3]) then provides an integer index ind(A) for each geometric class A. The classes with nonzero index are called essential Nielsen classes. The Nielsen number N (f ) of f is then the number of essential classes of f . Classical Nielsen theory establishes the important fact that the number N (f ) is a homotopy invariant lower bound for min{#(Φ(g)) | g  f } which is sharp in many cases. 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 basepoint 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. There is an induced homomorphism f∗ω : π1 (X, x0 ) → π1 (X, x0 ) by f∗ω (α) = ωf∗ (α)ω −1 . The map f∗ω 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∗ω (δ −1 ). The Reidemeister class containing α will be denoted by [α], and the set of all Reidemeister classes by R(f∗ω ) or R(f ) (by identification). The reader is encouraged to carefully distinguish the symbol R(f ) from the symbol R(f ) which denotes the Reidemeister number #R(f ) of f .

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The algebraic and geometric Nielsen classes are related by an injective function ρ : Φ (f ) → R(f∗ω ) defined as follows. Given x ∈ A we choose a path c from the basepoint x0 to x. We can then define ρ(A) = [cf (c−1 )ω −1 ]. This will be independent of the choice of x within A and c. The set of the geometric Nielsen classes for f can be regarded as a subset of R(f∗ω ) via ρ and we can define an index for Reidemeister classes of f so that the index of [α] is the same as the index of the geometric class (empty or not) that determines it (see for example [18]). Thus we can define the Nielsen number N (f ) of f either as the number of essential geometric classes or as the number of essential algebraic classes of f . We remind the reader the mod H version of Nielsen theory. If f, X, x, and ω are as above, and if H is a normal subgroup of π1 (X) which is invariant under f∗ , then by replacing “= 0” in the above definition by “∈ H”, we generalize the notions of Nielsen classes, essential Nielsen classes, Nielsen number, Reidemeister classes etc., to mod H Nielsen classes, essential mod H Nielsen classes, mod H Nielsen number (the number of essential mod H Nielsen classes), and mod H Reidemeister classes. We denote these by ΦH (f ), εH (f ), NH (f ) and RH (f∗ω ) (or RH (f ) by identification) respectively, then we get an injective function ρH : ΦH (f ) → RH (f∗ω ) (see [18], [10] or [14] etc.). Lemma 2.1. [18, 1.3] Let G : f  g : X → X be a homotopy, where f satisfies f∗ (H) ⊂ H (so g∗ (H) ⊂ H). Let x0 ∈ X and ω be a path in X from x0 to f (x0 ), and let x0 ∈ X and ω  be a path in X from x0 to g(x0 ).  Let u be a path from x0 to x0 , then we have an induced bijection μG : RH (f∗ω ) → RH (g∗ω ) which preserves index, by μG ([aH ]) = [u−1 aω(G, u)ω  −1 H ], where aH is the mod H equivalence class of the loop a, and (G, u) is the diagonal path defined by (G, u)(t) = G(u(t), t). Lemma 2.2. [18, 1.4] Let X, Y be spaces and let f : X → X, g : Y → Y and h : X → Y be maps so that the diagram f

X −−−−→ ⏐ ⏐ h

X ⏐ ⏐ h

g

Y −−−−→ Y is commutative. Suppose further that H and H  are normal subgroups of π1 (X) and π1 (Y ) respectively, and f∗ (H) ⊂ H, g∗ (H  ) ⊂ H  , h∗ (H) ⊂ H  . Let x0 ∈ X and ω be a path in X from x0 to f (x0 ). If σ = h(ω), then we have an induced function h∗ : RH (f∗ω ) → RH  (g∗σ ) such that h∗ (ΦH (f )) ⊂ ΦH  (g), by h∗ ([aH ]) = [h ◦ aH  ]. Corollary 2.3. [18, 1.5] Let X, Y, H and H  be as above. Suppose that h : X → Y and h : Y → X are maps such that h∗ (H) ⊂ H  , h∗ (H  ) ⊂ H. Let f = h ◦ h : X → X and g = h ◦ h : Y → Y . Then the diagrams f

X −−−−→ ⏐ ⏐ h g

X ⏐ ⏐ h

Y −−−−→ Y

f

and

X −−−−→  ⏐ h ⏐ g

X  ⏐  ⏐h

Y −−−−→ Y

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are commutative, hence h∗ : (RH (f ), ΦH (f )) → (RH  (g), ΦH  (g)), h∗ : (RH  (g), ΦH  (g)) → (RH (f ), ΦH (f )) are defined. Moreover h∗ ◦ h∗ and h∗ ◦ h∗ are both the identities, so h∗ is a bijection with inverse h∗ , both h∗ and h∗ preserve index. Next we recall some elements of fibre space theory and its interplay with Nielsen theory. In this paper p we will assume that all of our fibrations F → E → B are Hurewicz fibrations. Recall that this means there is a continuous path lifting function λ : W = {(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 . ¯ with f : E → E and f¯ : B → B, satisfying By a fibre preserving map of p we mean a pair of maps (f, f) ¯ ¯ ¯ pf = f p. Similarly, a fibrewise homotopy (H, H) : (f, f )  (g, g¯) is a pair of homotopies H : E × I → E ¯ : B × I → B, satisfying pH = H(p ¯ × 1). If (f, f¯) is a fibre preserving map and b ∈ Φ(f¯) then we and H will use fb to denote the restriction f |p−1 (b) : p−1 (b) → p−1 (b) of f to the fibre. Let K denote the kernel of the inclusion induced homomorphism from π1 (F ) to π1 (E), then K is a normal invariant subgroupoid of π1 (F ). If (f, f¯) is a fibre preserving map and b ∈ Φ(f¯), then (fb )∗ (K) ⊂ K, so RK (fb ), ΦK (fb ) are defined. For each b ∈ B we denote the inclusion from p−1 (b) to E by ib . ¯ to (g, g¯). Let b ∈ Φ(f¯), b ∈ Φ(¯ ¯ be a fibrewise homotopy from (f, f) g) Lemma 2.4. [18, 2.2-2.5] Let (H, H)  ¯ ω ¯ be a path in B from b to b such that ω ¯  (H, ¯ ). Then there is an index-preserving bijection and ω Tω¯ : RK (fb ) → RK (gb ) such that the diagram T

RK (fb ) −−−ω¯−→ RK (gb ) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b ∗ R(f )

μH

−−−−→

R(g)

is commutative. Lemma 2.5. [18, 4.2] Let H : f  g : E × I → E be a fibrewise homotopy, and let F ∈ Φ (f ), F = μH (F). Then i(f¯; p∗ (F)) = i(¯ g ; p∗ (F )). Suppose further that p∗ (F ) ∈ Φ (¯ g ). Then for any b ∈ p∗ (F), any b ∈ p∗ (F ),  −1 −1  i(fb ; p (b) ∩ F) = i(gb ; p (b ) ∩ F ). Lemma 2.6. [18, 4.3] Let (E, p, B) and (E  , p , B  ) be fibre spaces with E, E  , B, B  and all fibres compact connected ANR’s. Let h : E → E  and h : E  → E be fibre preserving maps, and set f = h ◦ h, g = h ◦ h . Suppose F ∈ Φ (f ), F = h∗ (F). Then i(f¯; p∗ (F)) = i(¯ g ; p∗ (F )), and for any b ∈ p∗ (F), any b ∈ p∗ (F ), i(fb ; p−1 (b) ∩ F) = i(gb ; p−1 (b ) ∩ F ). Theorem 2.7 (Product formula for the index of a fixed point class of fibre preserving maps). [18, 4.1] Let p : E → B be a fibration of compact connected ANR’s, and let f : E → E be a fibre preserving map. Suppose F ∈ Φ (f ). Then for any b ∈ p∗ (F), i(f ; F) = i(f¯; p∗ (F)) · i(fb ; p−1 (b) ∩ F). Corollary 2.8. [18, 4.1] Suppose that p and f are as Theorem 2.7. Suppose F ∈ Φ (f ). Then i(f ; F) = 0 if and only if i(f¯; p∗ (F)) = 0 and for some (hence all) b ∈ p∗ (F), i(fb ; p−1 (b) ∩ F) = 0.

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2.2. Local Nielsen theory This subsection is a quick review of local Nielsen fixed point theory in [1]. Let U be an open subset in a metric ANR X. A map f : U → X is said to be admissible if (i) f is compactly fixed, i.e., Φ(f ) = {x ∈ ¯ ⊂ U and f |W is U | f (x) = x} is compact in U , and (ii) there exists an open W such that Φ(f ) ⊂ W ⊂ W ¯ compact, where W is the closure of W . Then the fixed point index of f is defined by i(f ; U ) := ind(f |W ; W ), where ind is the fixed point index given in [8], for compact maps. We say that x, y ∈ Φ(f ) are f Nielsen equivalent provided that there is a path c in U from x to y so that c  f (c) in X. This equivalence relation partitions Φ(f ) into Nielsen classes. Since X is an AN R, it is uniformly locally contractible. It follows that each Nielsen class is open in Φ(f ). Then the compactness of Φ(f ) implies that there are only finite many such classes. Definition 2.9. [1] A Nielsen class F is said to be essential if for any open set W ⊂ U with F = W ∩ Φ(f ), the index i(f ; F) := ind(f |W ; W ) is nonzero. The number of essential Nielsen classes of f is called the (local) Nielsen number of f , denoted by n(f ; U ). A homotopy H : U × I → X is said to be admissible if the map H : U × I → X × I given by H(x, t) = (H(x, t), t) is admissible. Theorem 2.10. [1] n(f ; U ) is invariant under admissible homotopy and is a lower bound for #Φ(g) for any g compactly fixed homotopic to f . 3. Fixed point classes of a locally defined map In this paper we consider Reidemeister classes of a locally defined map. We use a modified fundamental group approach as in [3,18], unlike [1] which used a covering space approach of Nielsen theory (see [1, p. 1964]). Let U be an open subset in a metric ANR X, and f : U → X be an admissible map. All the results in this section are similar to the corresponding results for fixed point class of a map f : X → X in [18, Section 1]. Let Φ (f ; U ) denote the set of local Nielsen classes of f , defined in [1]. Since the restriction of f to every connected component of U is also admissible, we may assume without loss of generality that both U and X are path connected. We choose a basepoint x0 ∈ U , along with a path ω from x0 to f (x0 ). Please note in what follows we shall not distinguish between a path and its path class in the fundamental groupoid of X. If x0 is a fixed point of f , we will not distinguish between x0 and the constant path at x0 . There are the induced homomorphisms i∗ : π1 (U, x0 ) → π1 (X, x0 ) and f∗ : π1 (U, x0 ) → π1 (X, f (x0 )) by the inclusion i : U → X and f respectively. We define a homomorphism f∗ω : π1 (U, x0 ) → π1 (X, x0 ) by f∗ω (α) = ωf∗ (α)ω −1 , so f∗ω (a) = ωf (a)ω −1 . Definition 3.1. Any two elements α, β of π1 (X, x0 ) are said to be “∼ equivalent”, denoted α ∼ β, provided that there is a γ ∈ π1 (U, x0 ) such that α = i∗ (γ)βf∗ω (γ −1 ). Since i∗ (π1 (U, x0 )) is a group, ∼ is an equivalence relation in π1 (X, x0 ). The set of all ∼ equivalence classes will be denoted by R(f∗ω , i∗ ), and the class containing α by [α]. Let Coin(f∗ω , i∗ ) = {α ∈ π1 (U, x0 ) | f∗ω (α) = i∗ (α)}.

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Then we have an exact sequence of based sets i∗ ·f −ω

j

∗ 0 → Coin(f∗ω , i∗ ) → π1 (U, x0 ) −−−− −→ π1 (X, x0 ) − → R(f∗ω , i∗ ) → 0

where the first function is inclusion, the second takes α to i∗ (α)f∗ω (α−1 ), and the third places an element β in it Reidemeister class [β]. If π1 (X, x0 ) is abelian, then i∗ · f∗−ω is a homomorphism which we then write additively as i∗ − f∗ω . In this case R(f∗ω , i∗ ) inherits a canonical abelian group structure as the cokernel of i∗ − f∗ω , and of course Coin(f∗ω , i∗ ) is the kernel of the same function. Thus we have Proposition 3.2. Let f : U → X be a map with π1 (X, x0 ) abelian, then the sequence i∗ −f ω

j

0 → Coin(f∗ω , i∗ ) → π1 (U, x0 ) −−−−∗→ π1 (X, x0 ) − → R(f∗ω , i∗ ) → 0 is an exact sequence of groups and homomorphisms. The proof of every result in this section is essentially the same as the proof of the corresponding result in ordinary theory (see [18]), so is omitted. Proposition 3.3. For each x ∈ Φ(f ), the set {cf (c−1 )ω −1  | for any path c in U from x0 to x} is exactly a ∼ equivalence class, so x determines an element of R(f∗ω , i∗ ). Two fixed points determine the same element of R(f∗ω , i∗ ) if and only if they are in the same Nielsen class. Definition 3.4. We define a function ρ(x0 , ω) : Φ (f ; U ) → R(f∗ω , i∗ ) as follows. Given F ∈ Φ (f ; U ), we choose a point x ∈ F and a path c in U from x0 to x, then define ρ(x0 , ω)(F) = [cf (c−1 )ω −1 ]. By Proposition 3.3, ρ(x0 , ω) is an injection. A pair (x, ω) with x ∈ U and ω a path from x to f (x) is called a reference pair for f . Proposition 3.5. Let (x, ω) and (x , ω  ) be two reference pairs for f , and u be a path in U from x to x .  Let ν : R(f∗ω , i∗ ) → R(f∗ω , i∗ ) be the transformation defined by ν([a]) = [u−1 aωf (u)ω  −1 ], then ν is independent of the choice of the loop a and the path u, and that ν is a bijection such that the diagram Φ (f ; U ) ⏐ ⏐ ρ(x,ω)

Φ (f ; U ) ⏐ ⏐   ρ(x ,ω ) ν



R(f∗ω , i∗ ) −−−−→ R(f∗ω , i∗ ) is commutative, where “=” denotes the identity. From this proposition, we can identity all R(f∗ω , i∗ ) by ν to get an abstract set denoted by R(f ; U ), and we can identify all ρ(x, ω) to get an injection ρ : Φ (f ; U ) → R(f ; U ). The elements of R(f ; U ) are called fixed point classes of f . Each element of R(f∗ω , i∗ ) is called the representation of the corresponding element of R(f ; U ) in the reference pair (x, ω). We will identify each element F of Φ (f ; U ) with ρ(F), and think of Φ (f ; U ) as a subset of R(f ; U ). Definition 3.6. An algebraic class [α] ∈ R(f ; U ) is said to be nonempty if it lies in the image of ρ. For any F ∈ R(f ; U ) we define its index I(f ; F) as follows: if F ∈ Φ (f ; U ), then I(f ; F) is the usual fixed point index (as in [8]), otherwise, I(f ; F) = 0. A fixed point class F ∈ R(f ; U ) is said to be essential provided it has nonzero index.

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Thus, we get that the Nielsen number n(f ; U ) is equal to the number of essential fixed point classes of f . The following example illustrates how the Reidemeister classes can be used to calculate the local Nielsen number n(f ; U ). Example 3.7. Let X = T 2 = S 1 ×S 1 = {(eθi , eφi ) | 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ 2π}, let S11 = {(eθi , 1) : 0 ≤ θ ≤ 2π} and S21 = {(1, eφi ) : 0 ≤ φ ≤ 2π} be the two circles in X, let A = S11 ∨ S21 , and let x0 = (1, 1) be the common point of the two circles. Let U = X − {q} be the punctured torus where q ∈ X − A. Define a map f0 : A → A by f0 (eθi , 1) = (e3θi , 1) on S11 , and by f0 (1, eφi ) = (1, eφi ) on S21 . There is a retraction r : U → A. Let f = i ◦ f0 ◦ r : U → X, where i : A → X is the inclusion. Now N (f0 ) = 2 with a Nielsen class containing the single point x1 = (−1, 1) in S11 , the other being the whole of the circle S21 (see [14, p. 90]). We shall use the injectivity of the function ρ : Φ (f ; U ) → R(f ; U ) (Proposition 3.3) to show that the Nielsen classes of f0 remain distinct when considered as fixed point classes of f so that n(f ; U ) = 2. We set the base point for U to be x0 = (1, 1) = (e0i , e0i ), and the path ω to be the constant path at x0 , then π1 (U, x0 ) ∼ = π1 (A, x0 ) = Z ∗ Z, where ∗ denotes free product of groups, and π1 (X, x0 ) = Z × Z. Let α and β be the loops given by α(t) = (e2πti , 1) and β(t) = (1, e2πti ) respectively, then α and β are generated π1 (U ) as well as π1 (X). We see that (i∗ − f∗x0 )(α) = α · α−3 = α−2 , (i∗ − f∗x0 )(β) = β · β −1 = 0, so Im(i∗ − f∗x0 ) is the infinite cyclic group with α2 as a generator. Let F 0 and F 1 be the Nielsen classes of f containing x0 and x1 respectively. We have that ρ(F 0 ) = [0], and ρ(F 1 ) = [cf (c−1 )ω −1 ] = [α] by taking the path c from x0 to x1 which is defined by c(t) = (eπti , 1). Since [0] and [α] are distinct elements of R(f∗x0 , i∗ ), then by Proposition 3.3, x0 and x1 are not in the same Nielsen class of f , thus F 0 and F 1 are distinct Nielsen classes of f . From the restrictivity of the index, we also get that i(f ; F 0 ) = i(f0 ; F 0 ) = −2, and i(f ; F 1 ) = i(f0 ; F 1 ) = −1 as needed (see [14, p. 16, Corollary 3.7]). Let H : f  g : U → X be an admissible homotopy. The following proposition shows the relation between R(f ; U ) and R(g; U ). Proposition 3.8. Let (x, ω) be a reference pair for f , and (x , ω  ) be a reference pair for g. Let u be a  path in U from x to x . Let μH : R(f∗ω , i∗ ) → R(g∗ω , i∗ ) be the transformation defined by μH ([a]) = [u−1 aω(H, u)ω  −1 ]. The transformation μH : R(f ; U ) → R(g; U ) thus obtained does not depend on the pairs (x, ω), (x , ω  ) and the path u, and μH is a bijection. Let x0 ∈ F0 ∈ Φ (f ; U ) and x1 ∈ F1 ∈ Φ (g; U ). Then μH (F0 ) = F1 if and only if there is a path c in U from x0 to x1 such that c  (H, c). Thus μH preserves index. Suppose that X, Y are ANR’s and that h : X → Y is a map. Let V be a connected open subset in Y , and let U be a connected open subset in X such that h(U ) ⊂ V . Let f : U → X, g : V → Y be maps so that the diagram f

U −−−−→ ⏐ ⏐ h

X ⏐ ⏐ h

g

V −−−−→ Y is commutative. Then we have h(Φ(f )) ⊂ Φ(g). Furthermore, if both f and g are admissible, then we have Lemma 3.9. (a) If (x, ω) is a reference pair for f , then (h(x), h(ω)) is a reference pair for g. We can define h(ω) a transformation h∗ : R(f∗ω , i∗ ) → R(g∗ , i∗ ) by h∗ ([a]) = [h(a)]. Thus we get h∗ : R(f ; U ) → R(g; V ). (b) h∗ is independent of the choice of (x, ω).

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(c) If x ∈ F ∈ Φ (f ; U ), then h(x ) ∈ h∗ (F) ∈ Φ (g; V ). (d) If W is a connected open subset in another space Z, and k : Y → Z, l : W → Z are maps such that k(V ) ⊂ W, kg = lk, and l is admissible, then the diagram f

U −−−−→ ⏐ ⏐ h g

V −−−−→ ⏐ ⏐ k

X ⏐ ⏐ h Y ⏐ ⏐ k

l

W −−−−→ Z is commutative, and (kh)∗ = k∗ h∗ . Let X, X  be compact ANR’s, and U be an open subset in X. Suppose that h : U → X  , h : X  → X are maps, and U  = h −1 (U ). Set f = h h : U → X and g = hh : U  → X  . Suppose that Φ(f ) is compact. We can easily see that Φ(g) is also compact, and both f and g are admissible from compactness of X and X  . Then the diagram g

U  −−−−→ ⏐ ⏐ h 

X ⏐ ⏐  h

f

U −−−−→ X is commutative, and h∗ : (R(g; U  ), Φ (g; U  )) → (R(f ; U ), Φ (f ; U )) is defined. Then we have a slight generalization of the commutativity of the fixed point index in ordinary theory. Theorem 3.10. Suppose that F ∈ Φ (f ; U ). Then the union of the (nonempty) elements of h∗−1 (F) is h(F), and i(f ; F) =



i(g; G).

G∈h∗−1 (F)

Proof. We first prove h(F) is an union of Nielsen classes of g. It suffices to see that if x ∈ h(F), and x, y are g Nielsen equivalent, then y ∈ h(F). Let x ∈ h(F) and y be g Nielsen equivalent. By (c) of Lemma 3.9, h (x), h (y) are f Nielsen equivalent. Since h (x) ∈ h h(F) = F, h (y) ∈ F, so y = hh (y) ∈ h(F). We now prove that the union of the elements of h∗−1 (F) is h(F). Again by (c) of Lemma 3.9, for any G ∈ h∗−1 (F), any x ∈ G, h (x) ∈ h∗ (G) = F. This implies that h (G) ⊂ F, so G = hh (G) ⊂ h(F). Hence the union of the elements of h∗−1 (F) is contained in h(F). Conversely, if G ∈ Φ (g) is such that G ⊂ h(F), then for any x ∈ G, h (x) ∈ h h(F) = F. By (c) of Lemma 3.9, we also get h (x) ∈ h∗ (G). This implies that h∗ (G) = F, so G ⊂ h∗−1 (F). Hence h(F) is contained in the union of the elements of h∗−1 (F) as needed. Finally, by commutativity and additivity of the fixed point index, we get i(f ; F) = i(g; h(F)) =



i(g; G).

2

G∈h∗−1 (F)

4. Fixed point classes of a fibre preserving map defined locally Let (E, p, B) be a fibration in which E, B and all fibres F are compact connected ANR’s. Let V be an open subset in B, and U = p−1 (V ). Let p : U → V denote the restriction of p : E → B to U . We say that a

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9

map f : U → E is a fibre preserving map of p on U provided it induces a well defined base map f¯ : V → B with pf = f¯p (i.e., f¯ so that the diagram f

U −−−−→ ⏐ ⏐ p

E ⏐ ⏐p 

f¯

V −−−−→ B ¯ (or occasionally just f ) as a fibre preserving map on U . is commutative). We will refer to the pair (f, f) ¯ ¯ Similarly, a fibrewise homotopy (H, H) : (f, f )  (g, g¯) on U is a pair of homotopies H : U × I → E ¯ of p on U is said to be ¯ × id). A self fibre preserving map (f, f) ¯ : V × I → B, satisfying pH = H(p and H admissible if f¯ is compactly fixed, i.e., Φ(f¯) = {b ∈ V | f¯(b) = b} is compact in U . Then we see that f is also ¯ : V ×I → B×I ¯ on U is said to be admissible if the map H compactly fixed. A fibrewise homotopy (H, H) ¯ ¯ given by H(x, t) = (H(x, t), t) is compactly fixed. Then it follows that H : U × I → E × I is also compactly fixed. ¯ is an admissible fibre preserving map of p on Throughout this section, we will always assume that (f, f) U . As in Section 3, we may assume without loss of generality that V (and hence also U ) is path connected (note that since the subfibration p : U → V is a quotient map and all fibres are connected, then if V is connected then so is U ). For any b ∈ Φ(f¯) we denote by fb the map f restricted to p−1 (b). Let b ∈ Φ(f¯), let Φ (fb ) denote the set of ordinary Nielsen classes of fb : p−1 (b) → p−1 (b), and let R(fb ) denote the set of Reidemeister classes of fb (cf. [18, Section 1]). From Lemma 3.9 and the commutative diagram fb

p−1 (b) −−−−→ p−1 (b) ⏐ ⏐ ⏐i ⏐ ib  b U

f

−−−−→

E

we have Lemma 4.1. (a) If (x, ω) is a reference pair for fb , then (x, ω) is also a reference pair for f . We can define a ω transformation (ib )∗ : R(fb∗ ) → R(f∗ω , i∗ ) by (ib )∗ ([a]) = [ib (a)]. Thus we get (ib )∗ : R(fb ) → R(f ; U ). (b) (ib )∗ is independent of the choice of (x, ω). (c) If x ∈ F0 ∈ Φ (fb ), then x ∈ (ib )∗ (F0 ) ∈ Φ (f ; U ). ¯ be an admissible fibrewise homotopy on U from (f, f¯) : p → p to (g, g¯) : p → p. Let Lemma 4.2. Let (H, H)  ¯ ¯ ω b ∈ Φ(f ), b ∈ Φ(¯ g ) and ω ¯ be a path in V from b to b such that ω ¯  (H, ¯ ). Then there is an index-preserving bijection Tω¯ : R(fb ) → R(gb ) such that the diagram T

R(fb ) −−−ω¯−→ R(gb ) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b ∗ μH

R(f ; U ) −−−−→ R(g; U ) is commutative, where μH is the transformation given in Proposition 3.8. Proof. We indeed adopt the method used in [18] (cf. p. 223, Lemma 2.2). Define maps −1 h = τ(H,¯ (b) −→ p−1 (b), ¯ ω )−1 ◦ gb ◦ τω ¯ :p

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C. Li / Topology and its Applications 267 (2019) 106895

−1  h = τω¯ ◦ τ(H,¯ (b ) −→ p−1 (b ), ¯ ω )−1 ◦ gb : p −1  (b ) −→ p−1 (b ). h = τ(H,¯ ¯ ω ) ◦ τ(H,¯ ¯ ω )−1 ◦ gb : p

The maps h and h are a pair of commuting maps. For a path c, let csr (r, s ∈ I) denote the subpath of c defined ¯ ω by csr (t) = c(r + (s − r)t). Let D = {dt } : ω ¯  (H, ¯ ). We construct homotopies G : p−1 (b) × I → p−1 (b)   −1  −1  and G , G : p (b ) × I → p (b ) by G(x, t) = τ(H,¯ ¯ ))(t)), ¯ ω )0 ((H, λ(x, ω t  G (x , t) = τdt ◦ τ(H,¯ ¯ ω )−1 ◦ gb (x ),  ¯ ω ¯ )−1 )(1 − t)). G (x , t) = τ(H,¯ ¯ ω )1 (λ(gb (x ), (H, t

Then G is from fb to h, G is from h to h and G is from h to gb . By [18, Lemma 1.3], we have bijections μG , μG and μG . Let Tω¯ = μG ◦ μG ◦ (τω¯ )∗ ◦ μG : R(fb ) −→ R(gb ), then Tω¯ is a bijection and Tω¯ is index-preserving (see [18, p. 227]). As in [18], we can get an extremely simple representation of Tω¯ by a special choice of reference pairs. Thus let (x, r) be a pair for fb . The special choice for h is (x, rGx ) (cf. the remark following [18, 1.3]); the special choice for h is (x , τω¯ ◦ (rGx )), where x = τω¯ (x) (cf. [18, 1.4]); the special choice for h is (x , (τω¯ ◦ (rGx ))Gx ), and the special choice for gb is (x , (τω¯ ◦ (rGx ))Gx Gx ), where Gx , Gx , Gx are traces. Let r = (τω¯ ◦ (rGx ))Gx Gx . Then (x , r ) is called the induced pair from (x, r) and ω ¯ . As in [18],  r Tω¯ : R(fb∗ ) → R(gbr ∗ ) is given by Tω¯ ([a]) = [τω¯ ◦ a]. ¯ ), then by [18, 2.3], r  ω −1 r(H, ω) in E. Note that r  in p−1 (b ) depend on D, so the Let ω = λ(x, ω function Tω¯ may depend on the choice of the homotopy D used in its construction. We need only to prove the diagram 

T

r R(fb∗ ) −−−ω¯−→ R(gbr ∗ ) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b ∗ μH



R(f∗r , i∗ ) −−−−→ R(g∗r , i∗ ) is commutative. r ), From the representation of Tω¯ , Lemma 4.1 and [18, 2.1], we get that for any [a] ∈ R(fb∗ (ib )∗ ◦ Tω¯ ([a]) = (ib )∗ ([τω¯ ◦ a]) = [τω¯ ◦ a] = [ω −1 aω]. On the other hand, since ω is a path in U from x to x , by Lemma 4.1, Proposition 3.8 and [18, 2.3], μH ◦ (ib )∗ ([a]) = μH ([a]) = [ω −1 ar(H, ω)r −1 ] = [ω −1 aω].

2

¯ and the diagram in In the case when H is a constant homotopy ft ≡ f , b, b are in the same class of f, Lemma 4.2 becomes

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11

T

R(fb ) −−−ω¯−→ R(fb ) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b ∗ R(f ; U )

R(f ; U )

and so N (fb ) = N (fb ). Remark 4.3. It is possible to consider the mod K version of Lemma 4.2 (i.e., the question of the existence of a bijection Tω¯ : RK (fb ) → RK (gb )), where K is the kernel of the inclusion induced homomorphism from π1 (F ) to π1 (E). When X is a space and H is a normal subgroup of π1 (X), the mod H Nielsen classes and the mod H Reidemeister classes of a map h : X → X are defined under the condition that h∗ (H) ⊂ H. However, the commutative diagram (fb )∗

π1 (p−1 (b)) −−−−→ π1 (p−1 (b)) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b∗ π1 (U )

f∗

−−−−→

π1 (E)

need not imply the inclusion (fb )∗ (K) ⊂ K, so mod K Nielsen theory for fb is not defined in general. This is the reason why we do not use the mod K Nielsen theory in this paper. In order to consider the mod K version of Lemma 4.2, additional conditions for π1 (U ) or K (for example, the conditions that the inclusion induced homomorphism i∗ : π1 (U ) → π1 (X) is injective, or K = 0, or K = π1 (F )), which imply the inclusion (fb )∗ (K) ⊂ K for any b ∈ Φ(f¯), will be needed. 5. Product formula for the index of a fixed point class of a fibre preserving map f : U → E Let (E, p, B) be a fibration in which E, B and all fibres F are compact connected ANR’s. Let V be ¯ is an an open subset in B, and U = p−1 (V ). Throughout this section, we will always assume that (f, f) admissible fibre preserving map of p on U . In this section we prove the following main theorem: Theorem 5.1 (Product formula for the index of a fixed point class). Suppose F ∈ Φ (f ; U ). Then for any b ∈ p∗ (F), i(f ; F) = i(f¯; p∗ (F)) · i(fb ; p−1 (b) ∩ F). This theorem is an analogue of [18, Theorem 4.1]. Cheng Ye You in [18] proved the same equality first for the fibre preserving map f : E → E of a fibre space (E, p, B) with the base B a finite polyhedron, and then he used Borsuk Domination Theorem to extend the result to the case that E, B and all fibres are compact connected ANRs. Here we follow this method, however, in the case of local fibre preserving map f : U → E there is no natural bijection between the sets of Reidemeister classes, induced by a domination. Thus to get the product formula we have to use more direct arguments, which makes the proof much longer (see the proof of Theorem 5.1). To prove Theorem 5.1, we first give several lemmas. Lemma 5.2. Let H : f  g : U × I → E be an admissible fibrewise homotopy on U , and let F ∈ R(f ; U ), F = μH (F). Then i(f¯; p∗ (F)) = i(¯ g ; p∗ (F )). Suppose further that p∗ (F) ∈ Φ (f¯; V ), and p∗ (F ) ∈ Φ (¯ g ; V ).   Then for any b ∈ p∗ (F), any b ∈ p∗ (F ), i(fb ; p−1 (b) ∩ F) = i(gb ; p−1 (b ) ∩ F ).

12

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¯ : f¯  g¯ : V × I → B be the homotopy induced by H. It is easy to check that the diagram Proof. Let H μH

R(f ; U ) −−−−→ R(g; U ) ⏐ ⏐ ⏐p∗ ⏐ p∗   μ¯

R(f¯; V ) −−−H−→ R(¯ g; V ) is commutative. Then μH¯ (p∗ (F)) = p∗ (μH (F)) = p∗ (F  ), so i(f¯; p∗ (F)) = i(¯ g ; p∗ (F  )) (because μH¯ preserves     index). If p∗ (F ) ∈ Φ (¯ g ; V ), we take b ∈ p∗ (F) and b ∈ p∗ (F ). By Proposition 3.8, there is a path ω ¯ in V   −1 −1 ¯ from b to b such that ω ¯  (H, ω ¯ ). By Lemma 4.2, Tω¯ ((ib )∗ (F)) = (ib )∗ (F ). Since Tω¯ is a bijection and preserves index, we get i(fb ; p−1 (b) ∩ F) =





i(fb ; G) =

G∈(ib )−1 ∗ (F)

i(gb ; G ) = i(gb ; p−1 (b ) ∩ F  ),

 G ∈(ib )−1 ∗ (F )

where the first and the last equalities follow from the additivity of the fixed point index. 2 In the case when H is a constant homotopy ft ≡ f , b, b are in the same class of f¯, we have Corollary 5.3. Let F ∈ R(f ; U ) so that p∗ (F) ∈ Φ (f¯; V ). Then for any b ∈ p∗ (F), the index i(fb ; p−1 (b) ∩ F) is a constant. Let (E, p, B) and (E  , p , B  ) be fibre spaces with E, E  , B, B  and all fibres compact connected ANR’s. Let V be an open subset in B, and set U = p−1 (V ). Suppose that h : U → E  and h : E  → E ¯  : B  → B on the bases respectively. Set are fibre preserving maps which induce maps ¯h : V → B  , h   −1   −1   ¯ U = h (U ), V = h (V ). Let f = h h : U → E, g = hh |U  : U  → E  , then f, g are fibre preserving ¯h ¯  |V  : V  → B  on the bases respectively. Suppose that ¯ h ¯ : V → B, g¯ = h maps which induce maps f¯ = h the fibre preserving map f : U → E is admissible. It is easy to see that the fibre preserving map g : U  → E  is also admissible. Then we have

¯ Theorem 5.4. Suppose F ∈ Φ (f ; U ), and h∗−1 (F) = {G1 , ..., Gs }. Let b ∈ Φ(f¯), and b = h(b) ∈ Φ(¯ g ). Then i(fb ; p−1 (b) ∩ F) =

s  i(gb ; p −1 (b ) ∩ Gi ) = i(gb ; p −1 (b ) ∩ h(F)). i=1

¯  p , by (d) of Lemma 3.9, the diagram Proof. Since ph = h h

Φ (g; U  ) −−−∗−→ Φ (f ; U ) ⏐ ⏐ ⏐p∗ ⏐ p∗   ¯

h Φ (¯ g ; V  ) −−−∗−→ Φ (f¯; V )

is commutative. Case I: If b ∈ / p∗ (F) then for any Gi ∈ h∗−1 (F), b ∈ / p∗ (Gi ). If it were not, then there would be  −1   ¯  (b ) ∈ h ¯  p (Gi ) = Gi ∈ h∗ (F) with b ∈ p∗ (Gi ). From the commutativity of the diagram above, b = h ∗ ∗  −1  −1  −1  p∗ h∗ (Gi ) = p∗ (F), and this is a contradiction. Thus p (b)∩F = ∅, and for any Gi ∈ h∗ (F), p (b )∩Gi = ∅. Then we have i(fb ; p−1 (b) ∩ F) =

s  i(gb ; p −1 (b ) ∩ Gi ) = i(gb ; p −1 (b ) ∩ h(F)) = 0. i=1

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¯ ¯  (b ) = f¯(b) = b. Let Case II: Suppose b ∈ p∗ (F). Then b = h(b) and h hb = h|p−1 (b) : p−1 (b) → p −1 (b ), hb = h |p −1 (b ) : p −1 (b ) → p−1 (b). Then fb = hb hb and gb = hb hb . By [18, 1.5], we get bijections (hb )∗ : (R(fb ), Φ (fb )) → (R(gb ), Φ (gb )) and (hb )∗ : (R(gb ), Φ (gb )) → (R(fb ), Φ (fb )) such that both (hb )∗ and (hb )∗ preserve index, and (hb )∗ is the inverse of (hb )∗ . It is easy to check that by Lemma 3.9 and Lemma 4.1, the diagram (h  )∗

b R(gb ) −−− −→ R(fb ) ⏐ ⏐ ⏐(i ) ⏐ (ib )∗   b∗

h

R(g; U  ) −−−∗−→ R(f ; U )  −1 is commutative. Let (ib )−1 ∗ (F) = {C 1 , ..., C t }, and let D j = (hb )∗ (C j ) = (hb )∗ (C j ) ∈ R(gb ) for each −1 j = 1, ..., t. We have that for each C j , i(fb ; C j ) = i(gb ; D j ). From the diagram above, (hb )−1 ∗ ((ib )∗ (F)) = −1  −1 (ib )∗ (h∗ (F)), i.e.,  −1 (ib )−1 ∗ ({G 1 , ..., G s }) = (hb )∗ ({C 1 , ..., C t }) = {D 1 , ..., D t }.

By additivity of the fixed point index, we get i(fb ; p−1 (b) ∩ F) =

t t   i(fb ; C j ) = i(gb ; D j ) = j=1

=

s 

i(gb ; D)

D∈(ib )−1 ∗ ({G 1 ,...,G s })

j=1





i(gb ; D) =

i=1 D∈(i  )−1 b ∗ (G i )

s  i(gb ; p −1 (b ) ∩ Gi ) i=1

= i(gb ; p −1 (b ) ∩ (G1 ∪ · · · ∪ Gs )) = i(gb ; p −1 (b ) ∩ h(F)), where the last equality follows from the union G1 ∪ · · · ∪ Gs = h(F) of Theorem 3.10. 2 For A a subset of a metric space X, the diameter of A, denoted by diam(A), is the supremum of the set {d(x, y)|x, y ∈ A} where d is the metric of X. Two maps f, g : X → X are said to be -homotopic for  > 0 if there exists a map H : X × I → X such that H(x, 0) = f (x), H(x, 1) = g(x) for all x ∈ X, and diam(H(x × I)) <  for any x ∈ X. ϕ

ψ

A space Y dominates a space X by maps X −→ Y −→ X if ψ ◦ ϕ : X → X is homotopic to the identity ϕ ψ of X. A space Y -dominates a space X by maps X −→ Y −→ X if ψ ◦ ϕ is -homotopic to the identity of X (see [3, p. 40]). We say that a fibre space (E  , p , B  ) fibre dominates a fibre space (E, p, B) by fibre η

ξ

preserving maps E −→ E  −→ E if ξ ◦ η is fibrewise homotopic to the identity of E (see [18, p. 230]). To prove Theorem 5.1, we need a modified version of fibration domination theorem ([18, Lemma 4.5]): Lemma 5.5. Let (E, p, B) be a fibre space with E, B and all fibres compact connected ANR’s. Given  > 0, (E, p, B) is fibre dominated by another such fibre space (E  , p , B  ) with B  polyhedron by fibre preserving ξ η maps E −→ E  −→ E such that ξ¯◦ η¯ is -homotopic to the identity of B, where maps η¯ : B → B  , ξ¯ : B  → B are the maps induced by η : E → E  , ξ : E  → E on bases respectively.

C. Li / Topology and its Applications 267 (2019) 106895

14

η ¯

ξ¯

Proof. By [3, III.B.1], there exists a compact connected polyhedron B  which dominates B by B −→ B  −→ B such that ξ¯ ◦ η¯ is -homotopic to the identity of B. Then we can adopt the same way as in the proof of [18, 4.5] to obtain a fibre space (E  , p , B  ) which fibre dominates (E, p, B) by fibre preserving maps η ξ E −→ E  −→ E such that η : E → E  , ξ : E  → E induce η¯ : B → B  , ξ¯ : B  → B respectively. 2 The next lemma is an analogue of [3, VIII,A,2]. Lemma 5.6. Let X be a compact connected polyhedron, A be a subpolyhedron of X, and let f : A → X be a map. Given  > 0, f is ε-homotopic to a map f  : A → X such that f  has only a finite number of fixed points each lying in a maximal open simplex of some triangulation of X. Proof. Let d be the metric of X. Since X is a compact ANR, there exists  > 0 such that if f, g : A → X are maps and d(f, g) <  , where d(f, g) is the supremum of the set {d(f (x), g(x))|x ∈ A}, then f and g are -homotopic. In the same way as in the proof of Hopf’s Approximation Theorem (cf. [3, p. 118]), we can show that there exists a map f  : A → X such that f  has only a finite number of fixed points each lying in a maximal simplex of some triangulation of X, and d(f, f  ) <  . Then f  is a required map. 2 Proof of Theorem 5.1. Since Φ(f¯) is compact, there is an open subset V0 in B with Φ(f¯) ⊂ V0 ⊂ V¯0 ⊂ V . Let d be the metric of B. Since f¯ is fixed point free on ∂V0 , the number ε = inf{d(x, f¯(x)) | x ∈ ∂V0 } is positive. By Lemma 5.5, we get a fibre space (E  , p , B  ) with E  and all fibres compact connected ANR’s, and B  a compact connected polyhedron, which fibre dominates (E, p, B) by fibre preserving maps η ξ ¯ η¯ are induced maps by ξ, η E −→ E  −→ E such that ξ¯ ◦ η¯ is ε-homotopic to the identity of B, where ξ, on bases respectively. Let f1 = ξηf : U → E, then f1 is a fibre preserving map of p with induced map f¯1 = ξ¯η¯f¯ : V → B on the base. By the fibrewise homotopy from ξη to the identity of E, we get a fibrewise ¯ : (f, f¯)  (f1 , f¯1 ) on U such that H ¯ : f¯  f¯1 : V → B is ε-homotopy. By the choice of ε, homotopy (H, H) ¯ ¯ is compactly fixed on V0 × I. ¯ the fat homotopy H : V × I → B × I of H is fixed point free on ∂V0 × I, so H ¯ ¯ We will denote a fibre preserving map (f, f) with f : U → E and f : V → B by (f, f¯) : (U, V ) → (E, B), ¯ : (f, f¯)  (f1 , f¯1 ) with H : U × I → E and H ¯ : V × I → B by and denote a fibrewise homotopy (H, H) ¯ ¯ ¯ (H, H) : (f, f )  (f1 , f1 ) : (U, V ) → (E, B). Let U0 = p−1 (V0 ), and let (f, f¯) : (U0 , V0 ) → (E, B) and (f1 , f¯1 ) : (U0 , V0 ) → (E, B) be the restrictions ¯ : (f, f¯)  of (f, f¯) : (U, V ) → (E, B) and (f1 , f¯1 ) : (U, V ) → (E, B) to (U0 , V0 ) respectively, and (H, H) ¯ ¯ ¯ ¯ (f1 , f1 ) : (U0 , V0 ) → (E, B) be the restriction of (H, H) : (f, f )  (f1 , f1 ) : (U, V ) → (E, B) to (U0 ×I, V0 ×I). ¯ : (f, f¯)  Then (f, f¯), (f1 , f¯1 ) : (U0 , V0 ) → (E, B) are admissible fibre preserving maps of p, and (H, H) ¯ (f1 , f1 ) : (U0 , V0 ) → (E, B) is an admissible fibrewise homotopy. It is easy to check that the diagram μH

R(f ; U0 ) −−−−→ R(f1 ; U0 ) ⏐ ⏐ ⏐p∗ ⏐ p∗   μ¯

R(f¯; V0 ) −−−H−→ R(f¯1 ; V0 ) is commutative. Step 1. Clearly a Nielsen class of f on U splits into a disjoint union of Nielsen classes of f on U0 . Suppose ¯ = p∗ (F) ∈ Φ (f¯; V ), then F ¯ splits into a disjoint F = F 1 ∪ · · · ∪ F s with F 1 , ..., F s ∈ Φ (f ; U0 ). Let F ¯ =F ¯1 ∪ · · · ∪ F ¯ 1 , ..., F ¯ s ∈ Φ (f¯; V0 ). Note that for ¯ s with F union of Nielsen classes of f¯ on V0 . Suppose F ¯ each i = 1, ..., s, since by (c) of Lemma 3.9 p(F i ) ⊂ p∗ (F i ) ∈ Φ (f¯; V0 ) and p(F i ) ⊂ p(F) ⊂ p∗ (F) = F,  ¯ ¯ ¯ ¯ ¯ p∗ (F i ) ∩ F = ∅, this implies that p∗ (F i ) ⊂ F, so for some i (1 ≤ i ≤ s ), p∗ (F i ) = F¯i . ¯ i = μ ¯ (F ¯ i ) ∈ Φ (f¯1 ; V0 ) Let Gi = μH (F i ) ∈ Φ (f1 ; U0 ) for each i = 1, ..., s, and G = G1 ∪ · · · ∪ Gs . Let G H ¯ =G ¯1 ∪ · · · ∪ G ¯ s . From the diagram above, we get that for each i = 1, ..., s, for each i = 1, ..., s , and G

C. Li / Topology and its Applications 267 (2019) 106895

15

¯ ¯) = G ¯ ¯ ⊂ G, ¯ p∗ (Gi ) = p∗ ◦ μH (F i ) = μH¯ ◦ p∗ (F i ) = μH¯ (F i i so, ¯ p(G) = p(G1 ) ∪ · · · ∪ p(Gs ) ⊂ p∗ (G1 ) ∪ · · · ∪ p∗ (Gs ) ⊂ G. By additivity and homotopy invariance of the fixed point index, i(f ; F) =

s 

i(f ; F i ) =

i=1

i=1



¯ = i(f¯; F)

s 

s  i(f1 ; Gi ) = i(f1 ; G), 

¯ i) = i(f¯; F

i=1

s  ¯ i ) = i(f¯1 ; G). ¯ i(f¯1 ; G i=1

¯ any d ∈ G, ¯ i(fb ; p−1 (b) ∩ F) = i((f1 )d ; p−1 (d) ∩ G). Let d ∈ G ¯ k for some k, and We show that for any b ∈ F, −1 −1 ¯ choose b ∈ F k . We first prove that for each i = 1, ..., s, i((f1 )d , p (d) ∩ Gi ) = i(fb , p (b) ∩ F i ). Note that ¯ k , then from the diagram above, p∗ (F i ) = F ¯ k , so by p∗ : R(f ; U0 ) → R(f¯; V0 ) is not injective. If p∗ (Gi ) = G −1 −1 −1 ¯ ¯ Lemma 5.2, i((f1 )d ; p (d) ∩ Gi ) = i(fb ; p (b) ∩ F i ). If p∗ (Gi ) = Gk , then p∗ (F i ) = F k , so p (d) ∩ Gi = ∅ and p−1 (b) ∩ F i = ∅, thus i((f1 )d ; p−1 (d) ∩ Gi ) = i(fb ; p−1 (b) ∩ F i ) = 0. By additivity of the fixed point index, i((f1 )d ; p−1 (d) ∩ G) =

s 

i((f1 )d ; p−1 (d) ∩ Gi )

i=1

=

s 

i(fb ; p−1 (b) ∩ F i ) = i(fb ; p−1 (b) ∩ F).

i=1

¯ by Corollary 5.3, i((f1 )d ; p−1 (d) ∩G) Furthermore, since the index i(fb ; p−1 (b) ∩F) is independent of b ∈ F ¯ ¯ ¯ is independent of d ∈ G. Thus we have that for any b ∈ F, any d ∈ G, i(fb ; p−1 (b) ∩ F) = i((f1 )d ; p−1 (d) ∩ G). Step 2. Let V0 = ξ¯−1 (V0 ), and let U0 = p −1 (V0 ). We define f  = ηf ξ : U0 → E  , then f  is a fibre preserving map of p with induced map f¯ = η¯f¯ξ¯ : V0 → B  on the base. Since (f1 , f¯1 ) : (U0 , V0 ) → (E, B) is an admissible fibre preserving map, so is (f  , f¯ ) : (U0 , V0 ) → (E  , B  ). Without loss of generality, we can assume that G1 , ..., Gr (r ≤ s) are nonempty Nielsen classes so that G = G1 ∪ · · · ∪ Gr , and we can assume ¯ 1 , ..., G ¯ r (r ≤ s ) are nonempty classes so that G ¯ =G ¯1 ∪ · · · ∪ G ¯ r . that G   ¯ ¯ ¯ ¯ Let F = ηf (G), F = η¯f (G). Since p(G) ⊂ G, we get ¯ =F ¯ . p (F  ) = p ηf (G) = η¯f¯p(G) ⊂ η¯f¯(G) By commutativity of the fixed point index, i(f1 ; G) = i(f  ; ηf (G)) = i(f  ; F  ), ¯ = i(f¯ ; η¯f¯(G)) ¯ = i(f¯ ; F ¯  ). i(f¯1 ; G) ¯  be arbitrary and let ξ(b ¯ and η¯f¯(d) = b . By additivity of the fixed point index ¯  ) = d, then d ∈ G Let b ∈ F and Theorem 5.4, we get

16

C. Li / Topology and its Applications 267 (2019) 106895

i((f1 )d ; p−1 (d) ∩ G) =

r  i((f1 )d ; p−1 (d) ∩ Gi ) i=1

=

r  i(fb ; p −1 (b ) ∩ (ηf (Gi ))) i=1

= i(fb ; p −1 (b ) ∩ (ηf (G1 ∪ · · · ∪ Gr ))) = i(fb ; p −1 (b ) ∩ F  ). 

¯ i(f  ; p −1 (b ) ∩F  ) is independent of b ∈ F ¯. Furthermore, since i((f1 )d ; p−1 (d) ∩G) is independent of d ∈ G, b We observe now that there exists an open subset V1 in B  such that Φ(f¯ ) ⊂ V1 ⊂ V¯1 ⊂ V0 and V¯1 is a subpolyhedron of B  . Let ε = dist(Φ(f¯ ), B  − V0 ), it is easy to see that ε > 0. Taking a triangulation of B  of mesh less than ε /2, and uniting the interiors of all simplices that meet Φ(f¯ ) provides such an open subset V1 , then V¯1 is the union of all simplices that meet Φ(f¯ ). Let f¯ : V¯1 → B  be the restriction of f¯ : V0 → B  to V¯1 . Let ε = inf{d (f¯ (x), x) | x ∈ ∂V1 } > 0, where d is the metric of B  . By Lemma 5.6, f¯ : V¯1 → B  is ε -homotopic to a map f¯ : V¯1 → B  such that f¯ has only a finite number of fixed points each lying ¯  : f¯  f¯ : V¯  → B  be an ε -homotopy. in a maximal open simplex of some triangulation of B  . Let H 1      ¯ ¯ ¯ By the homotopy lifting property of (E , p , B ), H : f  f  : V¯1 → B  lifts to a fibrewise homotopy ¯  : V¯  ×I → B  ×I of H ¯  is fixed point H  : f   f  : p −1 (V¯1 ) → E  . By the choice of ε , the fat homotopy H 1 ¯  is compactly fixed on V  × I. Set U  = p −1 (V  ), and let (f  , f¯ ) : (U  , V  ) → (E  , B  ) free on ∂V1 × I, so H 1 1 1 1 1  ¯  and (f , f ) : (U1 , V1 ) → (E  , B  ) be the restrictions of (f  , f¯ ) : (p −1 (V¯1 ), V¯1 ) → (E  , B  ) and (f  , f¯ ) : ¯  ) : (f  , f¯ )  (f  , f¯ ) : (U  , V  ) → (E  , B  ) (p −1 (V¯1 ), V¯1 ) → (E  , B  ) to (U1 , V1 ) respectively, and (H  , H 1 1  ¯  ¯  ¯  −1 ¯  ¯  be the restriction of (H , H ) : (f , f )  (f , f ) : (p (V1 ), V1 ) → (E  , B  ) to (U1 × I, V1 × I). Then ¯  ) : (f  , f¯ )  (f  , f¯ ), (f  , f¯ ) : (U1 , V1 ) → (E  , B  ) are admissible fibre preserving maps of p , and (H  , H (f  , f¯ ) : (U1 , V1 ) → (E  , B  ) is an admissible fibrewise homotopy. Step 3. Each ηf (Gi ) splits into a disjoint union of Nielsen classes of f  : U0 → E  by Theorem 3.10 and a Nielsen class of f  on U0 splits into a disjoint union of Nielsen classes of f  on U1 . Hence each ηf (Gi ) splits ¯  split into a ¯ i ) and F into a disjoint union of Nielsen classes of f  : U1 → E  , so is F  . Similarly, each η¯f¯(G      disjoint union of Nielsen classes of f¯ : V1 → B  . Suppose that F = F 1 ∪ · · · ∪ F t with F 1 , ..., F t ∈ Φ (f  ; U1 ), ¯ = F ¯  , ..., F ¯   ∈ Φ (f¯ ; V  ). ¯ ∪ · · · ∪ F ¯   with F and that F 1 t 1 t 1 ¯  , p (F  ) ∩ F ¯  = ∅, this For each i = 1, ..., t, since p (F i ) ⊂ p∗ (F i ) ∈ Φ (f¯ ; V1 ) and p (F i ) ⊂ p (F  ) ⊂ F ∗ i   ¯ , so for some ¯i (1 ≤ ¯i ≤ t ), p (F  ) = F ¯ ¯. implies that p∗ (F i ) ⊂ F i i ∗       ¯  = μ ¯  (F ¯  ) ∈ Φ (f¯ ; V  ) Let F i = μH  (F i ) ∈ Φ (f ; U1 ) for each i = 1, ..., t, and F = F 1 ∪· ·· ∪F t . Let F i i 1 H ¯  = F ¯  ∪ · · · ∪ F ¯  . for each i = 1, ..., t , and F 1 t In the same way as in Step 1, we get that for each i = 1, ..., t, ¯ ¯ ) = F ¯ ¯ ⊂ F ¯  , p∗ (F i ) = p∗ ◦ μH  (F i ) = μH¯  ◦ p∗ (F i ) = μH¯  (F i i so, ¯  . p (F  ) = p (F 1 ) ∪ · · · ∪ p (F t ) ⊂ p∗ (F 1 ) ∪ · · · ∪ p∗ (F t ) ⊂ F Furthermore, in the same way as in Step 1, we get i(f  ; F  ) = i(f  ; F  ), ¯  ) = i(f¯ ; F ¯  ), i(f¯ ; F ¯  , any b ∈ F ¯  , i(f  ; p −1 (b ) ∩ F  ) = i(f  ; p −1 (b ) ∩ F  ). and for any b ∈ F b b

C. Li / Topology and its Applications 267 (2019) 106895

17

¯  = {b , ..., b }. Since p (F  ) ⊂ F ¯  , we get that Step 4. Let F n 1 

¯ ) = p −1 (b ) ∪ · · · ∪ p −1 (b ). F  ⊂ p −1 (F 1 n Hence, by additivity of the fixed point index and [14, IV.3.1], we have that i(f  ; F  ) =

n 

i(f  ; F  ∩ p −1 (bi ))

i=1

=

n 

i(f¯ ; bi ) · i(fbi ; F  ∩ p −1 (bi ))

i=1

= i(fb ; F  ∩ p −1 (b )) ·

n  i(f¯ ; bi ) i=1



¯ ) · i(f  ; F  ∩ p −1 (b )), = i(f¯ ; F b ¯  . where b ∈ F ¯ = i(f¯ ; F ¯  ), and for any b ∈ p∗ (F), any b ∈ F ¯  , From the steps above, i(f ; F) = i(f  ; F  ), i(f¯; F) i(fb , p−1 (b) ∩ F) = i(fb ; p −1 (b ) ∩ F  ), so for any b ∈ p∗ (F), i(f ; F) = i(f¯; p∗ (F)) · i(fb ; p−1 (b) ∩ F).

2

Corollary 5.7. Let (f, f¯) be an admissible fibre preserving map of p on U . Suppose F ∈ Φ (f ; U ). Then i(f ; F) = 0 if and only if i(f¯; p∗ (F)) = 0 and for some (hence all) b ∈ p∗ (F), i(fb ; p−1 (b) ∩ F) = 0. Proposition 5.8. If (f, f¯) is an admissible fibre preserving map of p on U , then n(f ; U ) ≤



N (fb ),

b∈ξ

¯ where ξ ⊆ Φ(f¯) consists of exactly one point from each essential fixed point class of f. ¯ 1 , · ··, F ¯ n be the essential fixed point classes of f¯ : V → B, where n = n(f¯; V ). By Corollary 5.7, Proof. Let F  ¯ i . Let if F ∈ Φ (f ; U ) is essential, then p∗ (F) is also essential, so for some i, p∗ (F) = F ¯ i }, 1 ≤ i ≤ n, ci = #{F ∈ Φ (f ; U ) | i(f ; F) = 0, p∗ (F) = F then n(f ; U ) =

n  ci . i=1

¯ i . If F ∈ Φ (f ; U ), i(f ; F) = 0 and p∗ (F) = F ¯ i, For each i, let bi be the unique element of ξ that represents F −1 then by Corollary 5.7, i(fbi ; p (bi ) ∩ F) = 0, so by additivity of the fixed point index, F contains at least one essential fixed point class of fbi . This implies that ci ≤ N (fbi ). Therefore, n(f ; U ) =

n n   ci ≤ N (fbi ). i=1

i=1

2

18

C. Li / Topology and its Applications 267 (2019) 106895

Example 5.9. Let p : E → B be the trivial circle bundle over the 2-torus. Namely, let B = T 2 = {(u, v) ∈ C ×C : |u| = 1, |v| = 1}, F = S 1 = {w ∈ C : |w| = 1}, and E = B ×F . Let S11 = {(eθi , 1) : 0 ≤ θ ≤ 2π} and S21 = {(1, eφi ) : 0 ≤ φ ≤ 2π} be the two circles in B, let A = S11 ∨S21 , and let b0 = (1, 1) be the common point of the two circles. Let V = B − {q} be the punctured torus where q ∈ B − A, and let U = p−1 (V ) = V × F . Define a map f0 : A × F → A × F by f0 (u, 1, w) = (u3 , 1, w−1 ) on S11 × F , and by f0 (1, v, w) = (1, v, vw−1 ) on S21 × F . Let r : V → A be a retraction. Let f = i ◦ f0 ◦ (r × id) : U → E with the identity id : F → F and the inclusion i : A × F → E. Then f is an admissible fibre preserving map on U inducing the map f¯ = ¯i ◦ f¯0 ◦ r : V → B, where f¯0 : A → A is given by f¯0 (u, 1) = (u3 , 1) on S11 , by f¯0 (1, v) = (1, v) on S21 , and ¯ 1 containing the single ¯i : A → B is the inclusion. Example 3.7 gives that n(f¯; V ) = 2 with a Nielsen class F 1 1 ¯ point b1 = (−1, 1) in S1 , the other F 0 being the whole of the circle S2 . The map on any fibre is a map of degree −1 so for any fixed point b in the base, N (fb ) = 2. We see that N (f0 ) = 3 with two Nielsen classes containing the single point x1 = (−1, 1, 1), and the single point x2 = (−1, 1, −1) (in fact, which are fixed point classes of fb1 ) respectively, the other containing both of fixed point classes of fb0 (see [14, p. 90]). The reader may wrongly presume, as in Example 3.7, that the Nielsen classes of f0 remain distinct when considered as fixed point classes of f so that n(f ; U ) = 3. The focus of our interest in this example is with two points x1 and x2 . It is here that the essence of the geometry takes place. We shall show that two Nielsen classes {x1 } and {x2 } of f0 coalesce into a single fixed point class of f , and n(f ; U ) = 2. From Corollary 5.7, an essential Nielsen class of f should contain an essential Nielsen class of fb0 or fb1 . As in the ordinary Nielsen theory for fibrations, it is easy to check from Lemma 3.9 that a fixed point class of fb0 and a fixed point class of fb1 can not coalesce into a single fixed point class of f because b0 and b1 are not in the same fixed point class of f¯. We set the base point for U to be x0 = (1, 1, 1) = (e0i , e0i , e0i ), and the path ω to be the constant path at x0 , then π1 (U, x0 ) ∼ = π1 (A × F, x0 ) = (Z ∗ Z) × Z, and π1 (E, x0 ) = Z × Z × Z. Let α, β and γ be the loops given by α(t) = (e2πti , 1, 1), β(t) = (1, e2πti , 1) and γ(t) = (1, 1, e2πti ) respectively, then α, β and γ generate π1 (U ) as well as π1 (E). We see that (i∗ − f∗x0 )(α) = α · α−3 = α−2 , (i∗ − f∗x0 )(β) = β · (βγ)−1 = γ −1 and (i∗ − f∗x0 )(γ) = γ 2 so Im(i∗ − f∗x0 ) is the free abelian group having {α2 , γ} as a basis. By taking the path c1 from x0 to x1 which is defined by c1 (t) = (eπti , 1, 1), and the path c2 from −1 x0 to x2 which is defined by c2 (t) = (eπti , 1, eπti ), we have that ρ({x1 }) = [c1 f (c−1 ] = [α], and 1 )ω −1 −1 ρ({x2 }) = [c2 f (c2 )ω ] = [αγ] = [α]. Since x1 and x2 determine the same element of R(f∗x0 , i∗ ), then by Proposition 3.3, two Nielsen classes {x1 } and {x2 } of fb1 are contained in a single fixed point class, F 1 say, of f as needed. On the other hand the path (1, e2πti , eπti ) for t ∈ [0, 1], is a path of fixed points connecting the Nielsen classes {(1, 1, 1)} and {(1, 1, −1)} of fb0 , so two Nielsen classes of fb0 coalesce into the single Nielsen class, F 0 say, of f . Thus we have that f has two distinct Nielsen classes F 0 , F 1 , and that i(fb0 ; p−1 (b0 ) ∩ F 0 ) = i(fb1 ; p−1 (b1 ) ∩ F 1 ) = 2 = 0. It follows from Theorem 5.1 that F 0 , F 1 are essential fixed point classes of f . We close with an example where π1 (E) is nonabelian. Example 5.10. Let K 2 denote the Klein bottle. We will think of K 2 as a quotient space of R2 , its universal covering space, under the equivalence classes defined by (s, t) ∼ (s + k, (−1)k t) and (s, t) ∼ (s, t + k) for any p k ∈ Z. The only Mostow fibration for K 2 has the form S 1 → K 2 → S 1 where p is projection on the first factor. Let V = S 1 − {0, 1} be the open subset in the base circle B = S 1 = I/[0 ∼ 1], and U = p−1 (V ). The correspondence (s, t) → (4s, t) for 0 < s < 1 induces a well defined, admissible fibre preserving map (f, f¯) ¯ does not extend to a (fibre preserving) map K 2 → K 2 . We get that Φ(f¯) = { 1 , 2 }, on U . Note that (f, f) 3 3 n(f¯; V ) = 2 with Nielsen classes { 13 } and { 23 }. The fibre map f 13 has degree −1, f 23 has degree 1, so N (f 13 ) = 2, N (f 23 ) = 0. We shall show that the classes x0 = [( 13 , 0)] and x1 = [( 13 , 12 )] of f 13 remain distinct when considered as fixed point classes of f . We set the base point for U to be x0 , and the path ω to be the constant path

C. Li / Topology and its Applications 267 (2019) 106895

19

at x0 , then π1 (U, x0 ) = Z = β, and π1 (E, x0 ) = α, β | α = βαβ, where α, β are the loops given by α(t) = [( 13 + t, 0)], β(t) = [( 13 , t)] respectively. We see that i∗ · f∗−x0 (β) = β 2 so Im(i∗ · f∗−x0 ) is the infinite cyclic group with β 2 as a generator. We have that ρ({x0 }) = [0], and ρ({x1 }) = [β] by taking the path c from x0 to x1 which is defined by c(t) = [( 13 , 12 t)]. Since β is not in Im(i∗ · f∗−x0 ), then [β] and [0] are distinct elements in R(f∗x0 , i∗ ), so by Proposition 3.3, x0 and x1 are not in the same fixed point class of f . From Corollary 5.7, it follows that {x0 } and {x1 } are essential fixed point classes of f , and the whole of the fibre p−1 ( 23 ) is an inessential fixed point classes of f , hence n(f ; U ) = 2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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