Nielsen zeta function for homeomorphisms of 3-manifolds

Alexander Fel’shtyn^{∗}

Abstract

We prove that the Nielsen zeta function is a rational function or a radical of a rational function for orientation preserving homeomorphisms of closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds.

0 Introduction

Before moving on the results of the paper, we briefly describe the few basic notions of Nielsen fixed point theory which will be used. We assume X to be a connected, compact polyhedron and f : X → X to be a continuous map. Let p : X̃ → X be the universal cover of X and f̃: X̃ → X̃ a lifting of f, ie. p ∘ f̃ = f ∘ p. Two liftings f̃ and f̃′ are called conjugate if there is a γ ∈ Γ ≅ π_{1}(X) such that f̃′ = γ ∘ f̃∘ γ^{−1}. The subset p(F ix(f̃)) ⊂ F ix(f) is called the fixed point class of f determined by the lifting class [f̃]. Two fixed points x_{0} and x_{1} of f belong to the same fixed point class iff there is a path c from x_{0} to x_{1} such that c ≅ f ∘c ( homotopy relative endpoints). This fact can be considered as an equivalent definition of a non-empty fixed point class. Every map f has only finitely many non-empty fixed point classes, each a compact subset of X.

A fixed point class is called essential if its index is nonzero. The number of lifting classes of f (and hence the number of fixed point classes, empty or not) is called the Reidemeister Number of f, denoted R(f). This is a positive integer or infinity. The number of essential fixed point classes is called the Nielsen number of f, denoted by N(f).

∗Part of this work was conducted during author stay in Forschungsinstitut für Mathematik, ETH Zürich

The Nielsen number is always finite. R(f) and N(f) are homotopy invariants. In the category of compact, connected polyhedra the Nielsen number of a map is, apart from in certain exceptional cases, equal to the least number of fixed points of maps with the same homotopy type as f.

Taking a dynamical point on view, we consider the iterates of f, and we may define several zeta functions connected with Nielsen fixed point theory (see [2, 3, 5]). The Nielsen zeta function of f is defined as power series:

N_{f}(z) := exp (∑_{n=1}^{∞} \frac{N(f^{n})}{n}z^{n}) .

The Nielsen zeta function N_{f}(z) is homotopy invariant. The function N_{f}(z) has a positive radius of convergence which has a sharp estimate in terms of the topological entropy of the map f [5].

In the main theorem of this article we prove that the Nielsen zeta function is a rational function or a radical of a rational function for orientation preserving homeomorphisms of special Haken or special Seifert 3-manifolds.

The author would like to thank M. Gromov, R. Hill, L. Potyagailo, V.G. Turaev and S. Wang for stimulating discussions. In particular, Turaev proposed the author in 1987 the conjecture that the Nielsen zeta function is a rational function or a radical of a rational function for homeomorphisms of Haken or Seifert manifolds. The author would like to thank the Max-Planck-Institute für Mathematik, Bonn for kind hospitality and support.

1 Periodic maps and homeomorphisms of hyperbolic manifolds

We prove in corollary 1 of this subsection that the Nielsen zeta function is a radical of rational function for any homeomorphism of a compact hyperbolic 3-manifold. Lemma 1 and corollary 1 play also important role in the proof of the main theorem of this section.

We denote N(f^{n}) by N_{n}. We shall say that f : X → X is a periodic map of period m, if f^{m} is the identity map id_{X} : X → X. Let μ(d), d ∈ N, be the Möbius function of number theory. As is known, it is given by the following equations: μ(d) = 0 if d is divisible by a square different from one ; μ(d) = (−1)^{k} if d is not divisible by a square different from one , where k denotes the number of prime divisors of d; μ(1) = 1.

We give the proof of the following key lemma for the completeness.

Lemma 1 [5] Let f be a periodic map of least period m of the connected compact polyhedron X . Then the Nielsen zeta function is equal to

N_{f}(z) = ∏_{d|m} ^{d} \sqrt{(1 − z^{d})^{−P (d)}},

where the product is taken over all divisors d of the period m, and P (d) is the integer

P (d) = ∑_{d_{1}|d} μ(d_{1})N_{d|d_{1}}.

Proof Since f^{m} = id, for each j, N_{j} = N_{m+j}. If (k, m) = 1, then there exist positive integers t and q such that kt = mq + 1. So (f^{k})^{t} = f^{kt} = f^{mq+1} = f^{mq}f = (f^{m})^{q}f = f. Consequently, N((f^{k})^{t}) = N(f). Let two fixed point x_{0} and x_{1} belong to the same fixed point class. Then there exists a path α from x_{0} to x_{1} such that α ∗ (f ∘ α)^{−1} ≃ 0. We have f(α ∗ f ∘ α)^{−1}) = (f ∘ α) ∗ (f^{2} ∘ α)^{−1} ≃ 0 and a product α ∗ (f ∘ α)^{−1} ∗ (f ∘ α) ∗ (f^{2} ∘ α)^{−1} = α ∗ (f^{2} ∘ α)^{−1} ≃ 0. It follows that α ∗ (f^{k} ∘ α)^{−1} ≃ 0 is derived by the iteration of this process. So x_{0} and x_{1} belong to the same fixed point class of f^{k}. If two point belong to the different fixed point classes f, then they belong to the different fixed point classes of f^{k} . So, each essential class( class with nonzero index) for f is an essential class for f^{k}; in addition , different essential classes for f are different essential classes for f^{k}. So N(f^{k}) ≥ N(f). Analogously, N(f) = N((f^{k})^{t}) ≥ N(f^{k}). Consequently , N(f) = N(f^{k}). One can prove completely analogously that N_{d} = N_{di}, if (i, m/d) =1, where d is a divisor of m. Using these series of equal Nielsen numbers, one can regroup the terms of the series in the exponential of the Nielsen zeta function so as to get logarithmic functions by adding and subtracting missing terms with necessary coefficient. We show how to do this first for period m = p^{l}, where p is a prime number . We have the following series of equal Nielsen numbers:

N_{1} = N_{k}, (k, p^{l}) = 1(i.e., no N_{ip}, N_{ip^{2}}, ....., N_{ip^{l}}, i = 1, 2, 3, ....),

N_{p} = N_{2p} = N_{3p} = ....... = N_{(p−1)p} = N_{(p+1)p} = ...(no N_{ip^{2}}, N_{ip^{3}}, ..., N_{ip^{l}}) etc.; finally,

N_{p^{l−1}} = N_{2p^{l−1}} = .....(no N_{ip^{l}})

and separately the number N_{p^{l}}. Further,

∑_{i=1}^{∞} \frac{N_{i}}{i} z^{i} = ∑_{i=1}^{∞} \frac{N_{1}}{i} z^{i} + ∑_{i=1}^{∞} \frac{(N_{p} − N_{1})}{p} \frac{z^{p}^{i}}{i} +
+ ∑_{i=1}^{∞} \frac{(N_{p^{2}} − (N_{p} − N_{1}) − N_{1})}{p^{2}} \frac{z^{p^{2}}^{i}}{i} + ...
+ ∑_{i=1}^{∞} \frac{(N_{p^{l}} − ... − (N_{p} − N_{1}) − N_{1})}{p^{l}} \frac{z^{p^{l}}^{i}}{i}
= −N_{1} · log(1 − z) + \frac{N_{1} − N_{p}}{p} · log(1 − z^{p}) +
+ \frac{N_{p} − N_{p^{2}}}{p^{2}} · log(1 − z^{p^{2}}) + ...
+ \frac{N_{p^{l−1}} − N_{p^{l}}}{p^{l}} · log(1 − z^{p^{l}}).

For an arbitrary period m , we get completely analogously,

N_{f}(z) = exp (∑_{i=1}^{∞} \frac{N(f^{i})}{i}z^{i})
= exp (∑_{d|m} ∑_{i=1}^{∞} \frac{P (d)}{d} · \frac{z^{d}^{i}}{i} )
= exp (∑_{d|m} \frac{P (d)}{d} · log(1 − z^{d}))
= ∏_{d|m} ^{d} \sqrt{(1 − z^{d})^{−P (d)}}

where the integers P (d) are calculated recursively by the formula

P (d) = N_{d} − ∑_{d_{1}|d;d_{1}≠d} P (d_{1}).

Moreover, if the last formula is rewritten in the form

N_{d} = ∑_{d_{1}|d} P (d_{1})

and one uses the Möbius Inversion law for real function in number theory, then

P (d) = ∑_{d_{1}|d} μ(d_{1}) · N_{d/d_{1}},

where μ(d_{1}) is the Möbius function in number theory. The lemma is proved.

Corollary 1 Let f : M^{n} → M^{n}, n ≥ 3 be a homeomorphism of a compact hyperbolic manifold. Then by Mostow rigidity theorem f is homotopic to periodic homeomorphism g. So lemma 1 applies and the Nielsen zeta function N_{f}(z) is equal to

N_{f}(z) = N_{g}(z) = ∏_{d|m} ^{d} \sqrt{(1 − z^{d})^{−P (d)}},

where the product is taken over all divisors d of the least period m of g, and P (d) is the integer P (d) = ∑_{d_{1}|d} μ(d_{1})N(g^{d|d_{1}}).

The proof of the following lemma is based on Thurston’s theory of homeomorphisms of surfaces [14].

Lemma 2 [5]

The Nielsen zeta function of a homeomorphism f of a compact surface F is either a rational function or the radical of a rational function.

Proof The case of a surface with χ(F) > 0 and case of torus were considered in [5]. If surface has χ(F) = 0 and F is not a torus then any homeomorphism is isotopic to periodic one (see [9]) and Nielsen zeta function is a radical of rational function by lemma 1 . In the case of a hyperbolic (χ(F) < 0) surface , according to Thurston’s classification theorem, the homeomorphism f is isotopic either to a periodic or a pseudo-Anosov, or a reducible homeomorphism. In the first case the assertion of the lemma follows from lemma 1. If f is an pseudo-Anosov homeomorphism of a compact surface then for each n > 0, N(f^{n}) = F (f^{n}) [15]. Consequently, in this case the Nielsen zeta function coincides with the Artin-Mazur zeta function: N_{f}(z) = F_{f}(z). Since in [1] Markov partitions are constructed for a pseudo-Anosov homeomorphism, Manning’s proof [11] of the rationality of the Artin-Mazur zeta function for diffeomorphisms satisfying Smale’s axiom A carries over to the case of pseudo-Anosov homeomorphisms. Thus , the Nielsen zeta function N_{f}(z) is also rational. Now if f is isotopic to a reduced homeomorphism ϕ, then there exists a reducing system S of disjoint circles S_{1}, S_{2}, ..., S_{m} on intF such that

1) each circle S_{i} does not bound a disk in F;

2) S_{i} is not isotopic to S_{j}, i ≠ j;

3) the system of circles S is invariant with respect to ϕ;

4) the system S has an open ϕ-invariant tubular neighborhood η(S) such that each ϕ -component Γ_{j} of the set F − η(S) is mapped into itself by some iterate ϕ^{n_{j}}, n_{j} > 0 of the map ϕ; here ϕ^{n_{j}} on Γ_{j} is either a pseudo-Anosov or a periodic homeomorphism;

5) each band η(S_{i}) is mapped into itself by some iterate ϕ^{m_{i}}, m_{i} > 0; here ϕ^{m_{i}} on η(S_{i}) is a generalized twist (possibly trivial).

Since the band η(S_{i}) is homotopically equivalent to the circle S^{1} the Nielsen zeta function N_{ϕ_{i}^{m}}(z) is rational (see [5]). The zeta functions N_{ϕ}(z) and N_{ϕ_{i}^{m}}(z) are connected on the ϕ - component Γ_{j} by the formula N_{ϕ}(z) = n_{j} \sqrt{N_{ϕ_{j}^{n}}(z^{n_{j}})}; analogously, on the band η(S_{i}), N_{ϕ}(z) = ^{m_{j}} \sqrt{N_{ϕ_{j}^{m}}(z^{m_{j}})}. The fixed points of ϕ^{n}, belonging to different components Γ_{j} and bands η(S_{i}) are nonequivalent [7], so the Nielsen number N(ϕ^{n}) is equal to the sum of the Nielsen numbers N(ϕ^{n}/Γ_{j}) and N(ϕ^{n}/η(S_{i})) of ϕ -components and bands . Consequently, by the properties of the exponential, the Nielsen zeta function N_{ϕ}(z) = N_{f}(z) is equal to the product of the Nielsen zeta functions of the ϕcomponents Γ_{j} and the bands η(S_{i}), i.e. is the radical of a rational function.

2 Homeomorphisms of Seifert fibre spaces

Let M be a compact 3-dimensional Seifert fibre space. That is a space which is foliated by simple closed curves, called fibres, such that a fibre L has a neighborhood which is either a solid Klein bottle or a fibred solid torus T_{r}, where r denotes the number of times a fibre near L wraps around L. A fibre is regular if it has neighborhoods fibre equivalent to solid torus S^{1}×D^{2}, otherwise it is called a critical fibre. There is a natural quotient map p : M → F where F is a 2-dimensional orbifold, and hence, topologically a compact surface. The projection of the critical fibres, which we denote by S, consists of a finite set of points in the interior of F together with a finite subcollection of the boundary components. The reader is referred to [13] for details, for definitions about 2-dimensional orbifolds, their Euler characteristics and other properties of Seifert fibre spaces. An orbifold is hyperbolic if it has negative Euler characteristic. Hyperbolic orbifold admits hyperbolic structure with totally geodesic boundary.

Any fibre-preserving homeomorphism f : M → M naturally induces a relative surface homeomorphism of the pair (F, S), which we will denote by f̂. Recall from [13] that there is a unique orientable Seifert fibre space with orbifold P (2, 2) -projective plane with cone singular points of order 2, 2. We denote this manifold by M_{P (2,2)}.

Lemma 3 [13, 9] Suppose M is a compact orientable Seifert fiber space which is not T^{3}, S^{1} × D^{2}, T^{2} × I or M_{P (2,2)}. Then there is a Seifert fibration p : M → F , so that any orientation preserving homeomorphism on M is isotopic to a fiber preserving homeomorphism with respect to this fibration.

Observe that if M and F are both orientable, then M admits a coherent orientation of all of its fibres and the homeomorphism f either preserves fibre orientation of all the fibres or it reverses fibre orientation.

Lemma 4 [10] Let M be a compact, orientable aspherical, 3-dimensional Seifert fibre space such that the quotient orbifold F is orientable and all fibres have neighborhoods of type T_{1}. Let f : M → M be a fibre-preserving homeomorphism inducing f̂: F → F. If f preserves fibre orientation, then Nielsen number N(f) = 0. If f reverses fibre orientation, then N(f) = 2N(f̂).

Proof First, by small isotopy, arrange that f has a finite number of fibres which are mapped to themselves. If f preserves fibre orientation a further isotopy, which leaves f̂unchanged, ensures that none of these fibres contains a fixed point. Thus, Fix (f) = and so, N(f) = 0. If f reverses fibre orientation there are exactly 2 fixed points on each invariant fibre. Since M is aspherical lemma 3.2 in [13] ensures that for any invariant fibre the 2 fixed points on that fibre are in distinct fixed point classes. On the other hand, the restriction on the fibre types in the hypothesis implies that a Nielsen path in F can always be lifted to a Nielsen path path in M. In fact, there will be two distinct lifts of each path in F. As a result, f has two fixed point classes covering each fixed point class of f̂. Since the index of a fixed point class of f is the same as that of its projection under p the result follows.

Remark 1 As one can see lemma 4 deals with a restricted class of Seifert fibre space. M. Kelly gave some examples in [10] which indicate how critical fibres of Seifert fibre space effect Nielsen classes in the case when r > 1 in T_{r} and which difficulties arise in this case.

Definition 1 A special Seifert fibre space is a Seifert fiber space such that the quotient orbifold F is orientable and all fibres have neighborhoods of type T_{1}.

Theorem 1 Suppose M is a closed orientable aspherical manifold which is a special Seifert fibre space and f : M → M is an orientation preserving homeomorphism. Then the Nielsen zeta function N_{f}(z) is a rational function or a radical of a rational function.

Proof If f preserves fibre orientation, then f^{n} also preserves fibre orientation and so, by lemma 4 Nielsen numbers N(f^{n}) = 0 for all n and the Nielsen zeta function N_{f}(z) = 1. If f reverses fibre orientation , then f^{2} preserves fibre orientation, f^{3} reverses fibre orientation and so on. Thus, we have N(f^{2k+1}) = 2N(f̂^{2k+1}) and N(f^{2k}) = 0 for k = 0, 1, 2, ... . As result, the Nielsen zeta function N_{f}(z) equals

N_{f}(z) = exp (∑_{n=1}^{∞} \frac{N(f^{n})}{n}z^{n})
= exp (∑_{k=0}^{∞} \frac{N(f^{2k+1})}{2k + 1}z^{2k+1})

= exp (∑_{k=0}^{∞} \frac{2N(f̂^{2k+1})}{2k + 1}z^{2k+1})

= exp (2 · ∑_{n=1}^{∞} \frac{N(f̂^{n})}{n}z^{n} − 2 · ∑_{k=1}^{∞} \frac{N(f̂^{2k})}{2k}z^{2k})

= (Nf̂(z))^{2} · exp − (∑_{k=1}^{∞} \frac{N((f̂^{2})^{k})}{k}(z^{2})^{k})
= (Nf̂(z))^{2}/Nf̂^{2}(z^{2}).

From lemma 2 it follows that Nielsen zeta functions Nf̂(z) and Nf̂^{2}(z^{2}) are either rational functions or the radicals of rational functions. Consequently, the Nielsen zeta function N_{f}(z) is a rational function or a radical of a rational function.

3 Main theorem

Basic concepts about 3-manifold topology can be found in [8], in particular the Jaco-Shalen-Johannson decomposition of Haken manifolds and Seifert fiber spaces. See [14] for discussions on hyperbolic 3-manifolds. We recall some basic facts about compact connected orientable 3-manifolds. A 3-manifold M is irreducible if every embedded 2-sphere bounds an embedded 3-disk. By the sphere theorem [6], an irreducible 3-manifolds is a K(π, 1) Eilenberg-MacLane space if and only if it is a 3-disk or has infinite fundamental group.

A properly-embedded orientable connected surface in a 3-manifold is incompressible if it is not a 2-sphere and the inclusion induces a injection on the fundamental groups. An irreducible 3-manifold is Haken if it contains an embedded orientable incompressible surface. We use notation N(X) to denote a regular neighborhood of set X. A Haken 3-manifold M can be decomposed along a canonical set T of incompressible tori into pieces such that each component of M − N(T ) is either hyperbolic manifold, or twisted I-bundle over Klein bottle, or a Seifert fiber space with hyperbolic orbifold. This decomposition is called Jaco-Shalen-Johannson decomposition.

Definition 2 A special Haken manifold is a Haken manifold M such that each component of M −N(T ) in JSJ decomposition of M is either hyperbolic manifold, or a twisted I-bundle over Klein bottle, or a aspherical special Seifert fiber space with hyperbolic orbifold.

We need some definitions from the paper [9].

Definition 3 Suppose f : M → M is a map, and A, B are f-invariant sets of M. If there is a path γ from A to B such that γ ∼ f ∘ γ rel (A, B), then we say that A, B are f-related.

Definition 4 Suppose M is a compact 3-manifold with torus boundaries. A map f : M → M is standard on boundary if for any component T of ∂M, the map f/T is one of the following types: (1) a fixed point free map; (2) a periodic map with isolated fixed points; (3) a fiber preserving, fiber orientation reversing map with respect to some S^{1} fibration of T.

Definition 5 A map f on a compact 3-manifold M is said to have F Rproperty(fixed-point relating property) if the following is true: if A ∈ Fix(f) and B is either a fixed point of f or an f-invariant component of ∂M, and A, B are f-related by a path γ, then γ is A, B homotopic to a path in Fix(f).

Definition 6 A map f is a type I standard map if (1) f has F R-property,

(2) f is standard on boundary, (3) Fix(f) consists of isolated points, and (4) f is of flipped pseudo-anosov type at each fixed point. A map f is type II standard map if it satisfies (1), (2) above, as well as (3) Fix(f) is a properly embedded 1-dimensional submanifold and (4) f preserves a normal structure on Fix(f).

The main result of this section is the following theorem.

Theorem 2 Suppose M is a closed orientable manifold which is special Haken manifold, and f : M → M is an orientation preserving homeomorphism. Then the Nielsen zeta function N_{f}(z) is a rational function or a radical from rational function.

Proof Let T be (possibly empty) set of invariant tori of the JSJ decomposition of M . Then each component of M − IntN(T ) is either hyperbolic manifold, or a twisted I-bundle over Klein bottle, or a aspherical special Seifert fiber space with hyperbolic orbifold. Isotop f so that it maps N(T ) homeomorphically to itself. Suppose that P is a Seifert fibered component of M − IntN(T ) such that f(P) = P. By [13], f/P is isotopic to a fiber preserving map. Recall that a torus T in M is a vertical torus if it is union of fibers in M. By [9] lemma 1.10, we can find a set of vertical tori T^{∗} in P, cutting P into pieces which are either a twisted I-bundle over Klein bottle, or have hyperbolic orbifold, and a fiber preserving isotopy of f, so that after isotopy, the restriction of f on each invariant piece has periodic or pseudo-Anosov orbifold map. Adding all such T^{∗} to T , we get a collection of tori T′, such that : (1) f(N(T′) = N (T′); (2) each component M_{i} of M − IntN(T′) either ist a twisted I-bundle over Klein bottle or has hyperbolic orbifold; (3) If f maps a Seifert fibered component M_{i} to itself, and if M_{i} has hyperbolic orbifold, then f is fiber preserving, and the orbifold map f̂ on F_{i} is either periodic or pseudo-Anosov. By [9] we can isotop f so that restriction f/M_{i} is a standard map for all M_{i} and after that we can further isotop f, rel ∂N_{j}, on each component N_{j} of N(T′), so that it is a standard map on N_{j}. By the definition of standard maps, Fix(f) intersects each of M_{i} and N_{j} in points and 1-manifolds, so Fix(f) is a disjoint union of points, arcs, and circles. Jiang, Wang and Wu proved [9]theorem 9.1 that different components of Fix(f) are not equivalent in Nielsen sense. We repeat their proof here for the completeness. If exist two different components C_{0}, C_{1} of Fix(f) which are equivalent, then there is a path α connecting C_{0}, C_{1} such that f ∘ α ∼ α rel ∂. Denote by T′′ the set of tori ∂N(T′). Among all such α, choose one such that the number of components ♯(α − T′′) of the set α − T′′ is minimal. In below we will find another such curve α′′ with ♯(α′′ − T′′) < ♯(α − T′′), which would contradict the choice of α. Let D be a disk, and let h : D → M be a homotopy f ∘ α ∼ α rel ∂. We may assume that h is transverse to T′′, and ♯h^{−1}(T′′) is minimal among all such h. Then h^{−1}(T′′) consists of a properly embedded 1-manifold on D, together with possibly one or two isolated points mapped to the ends of α. T′′ is π_{1}-injective in M, so one can modify h to remove all circles in h^{−1}(T′′). Note that h^{−1}(T′′) must contain some arcs, otherwise α would lie in some M_{i} or N(T_{j}), which is impossible because the restriction of f in each piece has FRproperty. Now consider an outermost arc b in h^{−1}(T′′) . Let β = h(b). The ends of β can not both be on α, otherwise we can use the outermost disk to homotope α and reduce ♯(α − T′′), contradicting the choice of α. Since f is a homeomorphism, the same thing is true for f ∘ α. Therefore, β has one end on each of α and f · α. The arc b cutt off a disk Δ on D whose interior is disjoint from h^{−1}(T′′). The boundary of Δ gives rise to a loop h(∂Δ) = α_{1} ∪ β ∪ (f ∘ α_{1})^{−1}, where α_{1} is subpath of α starting from an end point x of α. let T be the torus in T′′ which contains β. Then the restriction of h on Δ gives a homotopy α_{1} ∼ f ·α_{1} rel(x, T). Since f has FR-property on each component of M −IntN(T ) and N(T ), by definition there is a path γ in Fix(f) such that γ ∼ α_{1} rel (x, T). Since γ is in Fix(f), the path α′ = γ^{−1} · α has the property that f ∘ α′ = (f ∘ γ^{−1}) · (f ∘ α) = γ^{−1} · (f ∘ α) ∼ γ^{−1} · α = α′ rel ∂. Since α_{1} ∼ γ rel (x, T), the path γ^{−1} · α_{1} is rel ∂ homotopic to a path δ on T. Write α = α_{1} · α_{2}. Then α′ = γ^{−1} · α = (γ^{−1} · α_{1}) · α_{2} ∼ δ · α_{2} rel ∂. By a small perturbation on δ, we get a path α′′ ∼ α′ rel ∂ such that ♯(α′′ −T′′) < ♯(α−T′′). Since f ∘α′′ ∼ f ∘α′ ∼ α′ ∼ α′′ rel ∂, this contradicts the minimality of ♯(α − T′′).

We can prove in the same way that the fixed points of an each iteration f^{n}, belonging to different components M_{i} and N_{j} are nonequivalent , so the Nielsen number N(f^{n}) equals to the sum of the Nielsen numbers N(f^{n}/M_{i}) and N(f^{n}/N_{j}) of components . Consequently, by the properties of the exponential, the Nielsen zeta function N_{f}(z) is equal to the product of the Nielsen zeta functions for the induced homeomorphisms of the components M_{i} and N_{j}. By corollary 1 the Nielsen zeta function of the hyperbolic component M_{i} is a radical of a rational function. The Nielsen zeta function is a radical of a rational function for the component M_{i} which is a aspherical special Seifert fibre space by the theorem 1 . The Nielsen zeta function is a rational function or a radical of a rational function for a homeomorphism of torus or Klein bottle by lemma 2 . This implies that the Nielsen zeta function is a rational function or a radical of a rational function for the induced homeomorphism of component N_{j} which is I-bundle over torus and for the induced homeomorphism of component M_{i} which is twisted I-bundle over Klein bottle. So, the Nielsen zeta function N_{f}(z) of the homeomorphism f of the whole manifold M is a rational function or a radical of a rational function as a product of the Nielsen zeta functions for the induced homeomorphisms of the components M_{i} and N_{j}.

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Institut für Mathematik, Ernst-Moritz-Arndt-Universität Greifswald Jahn-strasse 15a, D-17489 Greifswald, Germany. E-mail address: felshtyn@mail.uni-greifswald.de