J. Fixed Point Theory Appl. 12 (2012) 207–219 DOI 10.1007/s11784-012-0070-7 Published online February 2, 2012 © Springer Basel AG 2012

Journal of Fixed Point Theory and Applications

The mod H Nielsen Theory and the q-Nielsen Theory Nirattaya Khamsemanan Abstract. Let f : X → X be a self-map of a topological space. In 1992, M. Woo and J. Kim introduced the q-Nielsen theory which is a tool to estimate the Nielsen number. In this paper, we compare the qNielsen theory to the mod H Nielsen theory. We develop an algebraic formulation of the mod H theory which does not require the condition f (H) ⊂ H based on the universal covering space instead of the regular cover, as well as an algebraic formulation of the q-Nielsen theory. In fact, we show that the q-Nielsen theory is the same as the mod H theory for H = ker(qπ : π1 (X) → π1 (Y )), where qπ is the induced homomorphism of q and Y is a topological space. We also provide applications of the q-Nielsen theory and the mod H theory in the case where f (H) is not contained in H, as well as applications for wedge product spaces of nonaspherical types. Mathematics Subject Classification (2010). 55M20. Keywords. Nielsen theory, fixed point theory, minimum fixed point number, mod H Nielsen theory, q-Nielsen theory.

1. Introduction Let X be a topological space. Given a self-map f on X, it is quite difficult to calculate the Nielsen number directly from the definition alone. In cases where the Nielsen number N (f ) cannot be computed by existing methods, calculations of lower bounds for N (f ) are often of interest. In this paper, we investigate the q-Nielsen theory technique of Woo and Kim [18] and compare it with the mod H Nielsen theory [16, 7], another technique for estimating the Nielsen number. In Section 3, we discuss the mod H Nielsen theory and develop an algebraic definition of the mod H Reidemeister classes making use of a generalization of the theory. In Section 4, we discuss the work of Woo and Kim [18], regarding qNielsen theory, where q is a map from the space X to some space Y . In their

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work, two fixed points of f : X → X are q-equivalent if and only if there is a path c between them such that q ◦ c and q ◦ (f ◦ c) are homotopic. We also develop an equivalent definition for the q-Nielsen theory using covering spaces. In Section 5, we prove that, for H any normal subgroup of the fundamental group π1 (X), a space Y and a map q : X → Y can be constructed so that the q-Nielsen theory is the same as the mod H theory, where H is the kernel of the fundamental group homomorphism induced by q. However, q-Nielsen theory provides a different, geometric way to approach the mod H theory that makes it easier to prove that two fixed points are in the same mod H fixed point class.

2. Basic constructions  → X be the universal Let X be connected, compact polyhedron, and let p : X  to cover of X. For any self-map f : X → X, a lift f˜ of f is a map from X itself such that p ◦ f˜ = f ◦ p. A deck transformation is a homeomorphism  →X  such that p ◦ α = p. α:X The following statements are well-known theorems from the theory of covering spaces. (1) For any x0 ∈ X and x ¯0 , x ¯0 ∈ p−1 (x0 ), there exists a unique deck trans →X  such that α(¯ formation α : X x0 ) = x ¯0 . (2) Let x0 ∈ X and x1 = f (x0 ). Given x ¯0 ∈ p−1 (x0 ) and x ¯1 ∈ p−1 (x1 ), ˜ ˜ there exists a unique lift f of f such that f (¯ x0 ) = x ¯1 . It follows from (1) and (2) that for any two lifts f˜, f˜ of f , there exists a unique deck transformation α such that f˜ = α ◦ f˜. 2.1. The Nielsen fixed point theory Let f : X → X be a continuous map on X. Nielsen theory is concerned with finding the minimum number of solutions, denoted M F (f ), to the fixed point equation f (x) = x, minimized among all the maps in a given homotopy class. The set of all fixed points of the map f is denoted by Fix(f ). In Nielsen fixed point theory, fixed points x1 and x2 are equivalent if and only if there exists a path c from x1 to x2 such that c is homotopic to f ◦ c relative to the end points. The equivalence classes under this relation are called the Nielsen fixed point classes. The fixed point equivalence classes are classified as essential or not by means of the fixed point index of algebraic topology (see [2] for more details). The Nielsen number, denoted N (f ), is the number of Nielsen classes which have a nonzero index and it has the property that N (f ) ≤ M F (f ).

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2.2. Lifts  (see [4, p. 64]), the elements For a construction of universal covering spaces X  of X are in 1-1 correspondence with the end-point homotopy classes of paths  in X starting from x0 . For a given base point x0 ∈ X, let x ¯0 ∈ p−1 (x0 ) ⊂ X be the class of the constant path. Note that the class of x ¯0 is [e], the unit element in π1 (X, x0 ). An element  by α = [a] ∈ π1 (X, x0 ) acts on X α = [a] : [c] −→ α[c] = [a ∗ c]. The notation ∗ is used for the concatenation of two paths. Suppose f˜(¯ x0 ) = [w], where w is a path from x0 to f (x0 ). Since every lift f˜ is uniquely determined by its value f˜(¯ x0 ) ∈ p−1 (f (x0 )), then for any  we have point x ¯ = [c] ∈ X, f˜ : [c] −→ [w ∗ (f ◦ c)]. 2.3. Coordinate of the lifting class [β ◦ f˜] This section is a summary of the theory of the coordinate of the Nielsen lifting classes from [9]. We include some proofs because we will make use of the details later on. For a given base point x0 ∈ X, we can identify a specific deck transformation with an element of π1 (X, x0 ). Therefore, for a given lift f˜, any lift f˜ of f can be uniquely represented as α ◦ f˜, where α ∈ π1 (X, x0 ). For every α ∈ π1 (X, x0 ), the composition f˜ ◦ α is also a lift of f . Therefore, there is a unique element β ∈ π1 (X, x0 ) such that β ◦ f˜ = f˜ ◦ α. Definition 2.1. The endomorphism f˜π : π1 (X, x0 ) → π1 (X, x0 ) with respect to f˜ is defined by f˜π (α) ◦ f˜ = f˜ ◦ α. In fact, for every α = [a] ∈ π1 (X, x0 ), we have f˜π (α) = f˜π ([a]) = [w ∗ (f ◦ a) ∗ w−1 ], where w is a path from x0 to f (x0 ). Remark 2.1. If the base point x0 of π1 (X, x0 ) is a fixed point of f , then f˜π = fπ : π1 (X) → π1 (X). Definition 2.2. Two elements α and β of the fundamental group π1 (X, x0 ) are f˜π -conjugate if and only if there exists γ ∈ π1 (X, x0 ) such that β = γ ◦ α ◦ f˜π (γ −1 ). An f˜π -conjugacy class [α] is the set of all elements of the fundamental group that are f˜π -conjugate to α. A lifting class [α ◦ f˜] is the set of all lifts conjugated to α ◦ f˜; i.e., [α ◦ f˜] = {γ ◦ (α ◦ f˜) ◦ γ −1 | γ ∈ π1 (X, x0 )}.

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Lemma 2.1. For α, β ∈ π1 (X, x0 ), [α ◦ f˜] = [β ◦ f˜] if and only if α and β are f˜π -conjugate. Proof. [α ◦ f˜] = [β ◦ f˜] if and only if there exists γ ∈ π1 (X, x0 ) such that β ◦ f˜ = γ ◦ (α ◦ f˜) ◦ γ −1 = γ ◦ α ◦ f˜π (γ −1 ) ◦ f˜.



Definition 2.3. For a chosen base point x0 and a chosen lift f˜ of f , the f˜π conjugacy class of α ∈ π1 (X, x0 ) is called the coordinate of the class of a fixed point x if and only if x ∈ p(Fix(α ◦ f˜)). This α can be obtained geometrically. ¯0 ∈ Fix(f˜), where f˜ is the chosen Corollary 2.2. Suppose x0 ∈ p(Fix(f˜)) and x lift of f . Then the coordinate of the class of a fixed point x of f is the f˜π conjugacy class of α = [c ∗ (f ◦ c)−1 ] ∈ π1 (X, x0 ), where c is any path from x0 to x. In other words, x ∈ p(Fix(α ◦ f˜)).

3. The mod H theory Let H be a normal subgroup of π1 (X, x0 ). Like the classic Nielsen fixed point theory, there are two equivalent definitions of the mod H theory. Definition 3.1 (mod H fixed point classes (via path homotopy)). Two fixed points x1 and x2 of f : X → X are in the same mod H fixed point class if and only if there exists a path c from x1 to x2 such that [b ∗ c ∗ (f ◦ c)−1 ∗ b−1 ] ∈ H, where b is a path from x0 to x1 . Note that this definition is independent of the choice of b because H is a normal subgroup of π1 (X, x0 ). The second definition is via the lifting classes which we describe in the following subsection. 3.1. mod H lifting classes Unlike Jiang [9, p. 49], the following approach does not require that fπ (H) ⊆ H. Theorem 3.1 is the same as Jiang’s “mod H Nielsen theorem” which is Theorem 2.2 of [9] when fπ (H) ⊂ H.1 Definition 3.2. Two elements α and β in π1 (X, x0 ) are said to be mod H f˜π -conjugate if β = h ◦ γ ◦ α ◦ f˜π (γ −1 ) for some γ ∈ π1 (X, x0 ) and h ∈ H. Note that this is transitive, and hence an equivalence relation, because H is a normal subgroup. A mod H f˜π -conjugacy class [α]H is the set of all elements of the fundamental group that are mod H f˜π -conjugate to α. 1 The

presentation of the mod H Nielsen number in this subsection is due partly to the help from Seungwon Kim [15] and suggestions from the referee.

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Naturally, a mod H lifting class is an equivalent class in the set of all lifts of f to the universal cover; i.e.,  [α ◦ f˜]H = [β ◦ f˜]. β∈[α]H

Clearly, a mod H lifting class is a disjoint union of classical Nielsen lifting classes. Definition 3.3. The mod H coordinate of a fixed point x of f is the mod H f˜π -conjugacy class [a ∗ f (a−1 ) ∗ w−1 ]H , where a is a path from the base point x0 to x and w is the path from x0 to f (x0 ). Notice that this will give the homomorphism f˜π as described in the classical Nielsen theory. Definition 3.4 (mod H fixed point classes (via the lifting classes)). Suppose xi ∈ p(Fix(αi ◦ f˜)), i = 1, 2. Fixed points x1 and x2 are in the same mod H fixed point class if and only if [α1 ]H = [α2 ]H . Since H(= H(X, x0 )) is a normal subgroup of π1 (X, x0 ), for any path b from x0 to x1 , the induced map b∗ : π1 (X, x0 ) → π1 (X, x1 ) sends H(X, x0 ) onto H(X, x1 ), the image of H(X, x0 ) under the map b∗ . Hence in the case of the universal cover, the base point is not much of a concern. So for simplicity, we will prove the equivalence of the two definitions by using the base point x1 instead of x0 . For the rest of this subsection, the normal subgroup H refers to H(X, x1 ) in π1 (X, x1 ). Theorem 3.1. For any two fixed points x1 , x2 ∈ Fix(f ), the followings are equivalent. (1) The fixed points x1 , x2 are in the same mod H fixed point class (via lifting classes). (2) There exists a path c from x1 to x2 such that [c ∗ (f ◦ c)−1 ] ∈ H. Proof. (⇒) Suppose that x1 ∈ p(Fix(f˜)) and x2 ∈ p(Fix(h ◦ γ ◦ f˜ ◦ γ −1 )) for some h ∈ H and γ ∈ π1 (X, x1 ). Choose x ˜1 ∈ p−1 (x1 ) ∩ Fix(f˜) and −1 −1 ˜ x ˜2 ∈ p (x2 ) ∩ Fix(h ◦ γ ◦ f ◦ γ ). Since H is normal, there is h ∈ H such that h ◦ γ ◦ f˜ ◦ γ −1 = γ ◦ h ◦ f˜ ◦ γ −1 , x2 ) ∈ Fix(h ◦ f˜). and so x ˜2 ∈ Fix(γ ◦ h ◦ f˜ ◦ γ −1 ). This implies that γ −1 (˜ Now take a path c˜ from x˜1 to γ −1 (˜ x2 ). Then (h ◦ f˜◦ c˜)−1 is a path from −1  ˜ γ (˜ x2 ) to h (˜ x1 ) because x˜1 ∈ Fix(f ) and γ −1 (˜ x2 ) ∈ Fix(h ◦ f˜). Therefore,  −1  ˜ c˜(h ◦ f ◦ c˜) is a path from x ˜1 to h (˜ x1 ). Hence, c = p(˜ c) satisfies the condition that [c ∗ (f ◦ c)−1 ] = h ∈ H. (⇐) Suppose there is a path c from x1 to x2 such that [c∗(f ◦c)−1 ] ∈ H. For a chosen x ˜1 ∈ p−1 (x1 ), let c˜ be the lift of c starting at x ˜1 and ending at ˜ x ˜2 . Let f be the lift of f such that f˜(˜ x2 ) = x ˜2 . Then c˜ ∗ (f˜ ◦ c˜)−1 is a lift of

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the loop c ∗ (f ◦ c)−1 . Let x ˜1 ∈ p−1 (x1 ) be the end point of c˜ ∗ (f˜ ◦ c˜)−1 . Since −1 [c ∗ (f ◦ c) ] ∈ H, there is h ∈ H such that x ˜1 = h(˜ x1 ). This means that x ) = x ˜1 , h(f˜(˜ x1 )) = h(˜ 1

i.e., x ˜1 ∈ Fix(h ◦ f˜). Hence x1 ∈ p(Fix(h ◦ f˜)) and x2 ∈ p(Fix(f˜)) are in the same mod H fixed point class.  Jiang needed the condition fπ (H) ⊂ H because he used the fact that   f : X → X can be lifted to f˜H : X/H → X/H if and only if fπ (H) ⊂ H (see [9, p. 49]). Theorem 3.1 uses a different approach which does not require this condition. Definition 3.5. The mod H Reidemeister number of f , denoted RH (f ), is the number of mod H lifting classes. Example 1. Let T denote a torus whose fundamental group is π1 (T, x0 ) =

a, b | aba−1 b−1 . Suppose f : T → T is a self-map on a torus such that f induces fπ : π1 (T, x0 ) → π1 (T, x0 ), where fπ (a) = a−1 b2 , fπ (b) = ab−1 . We use the Fadell–Husseini formula in [3] to calculate the Reidemeister trace of f which is RT (f, f˜) = [1] − (−[a−1 ] − [ab−1 ]) + ([b−1 ] − [a−1 ] − [a−1 b]) = [1] + [b−1 ], where [ · ] denotes the f˜π -conjugacy class. Now consider the subgroup H generated by a in the fundamental group. Notice that H is a normal subgroup of π1 (T, x0 ). However, fπ (H) is not contained in H. Two mod H f˜π -conjugacy classes of f , [an1 bm1 ]H and [an2 bm2 ]H , are equal if and only if m1 ≡ m2 mod 2 (due to the fact that if γ = ar bt , then γ ◦ f˜π (γ −1 ) = a2r−t b2(t−r) ). Since [1]H = [b−1 ]H , the two f˜π -conjugacy classes [1] and [b−1 ] are not the same. Therefore N (f ) = 2.

4. The q-Nielsen theory In 1992, Woo and Kim [18] introduced another way to obtain a lower bound for the Nielsen number. Let q : X → Y be a map from the topological space X to a space Y and consider  X

f˜

p

 X

q

 Y

 /X p

f

 /X  Y

q

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Definition 4.1. Two fixed points x1 , x2 ∈ Fix(f ) are q-equivalent if there exits a path c from x1 to x2 such that q ◦ (f ◦ c) and q ◦ c are homotopic by a homotopy keeping the end points fixed. Remark 4.1. If two fixed points x1 , x2 ∈ Fix(f ) are in the same Nielsen fixed point class, then they are q-equivalent, where q is the identity map. Now we are going to develop an equivalent definition of the q-Nielsen classes via the universal covering space, where q is the identity map. We will choose the base point x0 of π1 (X, x0 ) so that x0 is a fixed point of f and choose f˜ so that x0 ∈ p(Fix(f˜)). Then for γ ∈ π1 (X, x0 ), an element of the fundamental group, f˜π (γ −1 ) = fπ (γ −1 ) = (fπ (γ))−1 . Definition 4.2. For a chosen base point x0 and a chosen lift f˜ of f , two lifts f˜ = α ◦ f˜ and f˜ = β ◦ f˜, with α, β ∈ π1 (X, x0 ), of f are q-equivalent if and only if there exists γ ∈ π1 (X, x0 ) such that qπ (β) = qπ (γ) ◦ qπ (α) ◦ qπ (f˜π (γ −1 )), where qπ : π1 (X, x0 ) → π1 (Y, q(x0 )) is the induced map. Remark 4.2. We call [f˜ ]q a q-lifting class where   [f˜ ]q = f˜ | f˜ is q-equivalent to f˜ . Suppose that f˜ = α ◦ f˜ and f˜ = β ◦ f˜ are in the same Nielsen lifting class, then we have [α ◦ f˜] = [β ◦ f˜]. From Lemma 2.1, there exists γ ∈ π1 (X, x0 ) such that β = γ ◦ α ◦ (fπ (γ))−1 . Therefore, qπ (β) = qπ (γ ◦ α ◦ (fπ (γ))−1 ); in other words, qπ (β) = qπ (γ) ◦ qπ (α) ◦ qπ (fπ (γ))−1 . Thus if f˜ , f˜ are in the same Nielsen lifting class, then they are in the same q-lifting class. We can conclude that  [f˜ ]q = [f˜ ]. f˜ ∈[f˜ ]q

Definition 4.3. A q-fixed point class, denoted Fq , is   Fq = x ∈ Fix(f ) | x ∈ p(Fix(f˜ )) for some f˜ ∈ [f˜ ]q . So, Fq =



p(Fix(f˜ )).

f˜ ∈[f˜ ]q

Theorem 4.1. Fixed points x1 and x2 are in the same q-fixed point class Fq if and only if x1 and x2 are q-equivalent.

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Proof. (⇐) Suppose that x1 and x2 are q-equivalent so there exists a path c from x1 to x2 such that q ◦ (f ◦ c)  q ◦ c. Let b be a path from x0 to x1 . Let β = [b∗(f ◦b)−1 ] and α = [b∗c∗(f (b∗c))−1 ] ∈ π1 (X, x0 ). From Corollary 2.2, x1 ∈ p(Fix(β ◦ f˜)) and x2 ∈ p(Fix(α ◦ f˜)). Thus we have qπ (α) = qπ ([b ∗ c ∗ (f ◦ (b ∗ c))−1 ]) = [q(b ∗ c ∗ (f ◦ (b ∗ c))−1 )] = [q(b) ∗ q(c) ∗ q(f (c−1 )) ∗ q(f (b−1 ))] = [q(b) ∗ q(f (b−1 ))] since q ◦ (f ◦ c)  q ◦ c = [q(b ∗ f (b−1 ))] = qπ ([b ∗ (f ◦ b−1 )]) = qπ (β). Therefore, β ◦ f˜ and α ◦ f˜ are in the same q-lifting class and so x1 , x2 ∈ Fq . (⇒) Suppose that x1 , x2 ∈ Fq . Let b be a path from x0 to x1 and let d be a path from x1 to x2 . From Corollary 2.2, x1 ∈ p(Fix(β ◦ f˜)) and x2 ∈ p(Fix(α ◦ f˜)), where β = [b ∗ (f ◦ b)−1 ] and α = [b ∗ d ∗ (f ◦ (b ∗ d))−1 ], respectively. Since x2 , x1 ∈ Fq , there exists γ = [w] ∈ π1 (X, x0 ) such that qπ (β) = qπ (γ) ◦ qπ (α) ◦ qπ (fπ (γ −1 )). Now, let c = b−1 ∗ w ∗ b ∗ d which is a path from x1 to x2 . Then we get qπ [b ∗ (c ∗ (f ◦ c)−1 ) ∗ b−1 ] = [q ∗ (b ∗ c ∗ (f ◦ c)−1 ∗ b−1 )] = [q(b) ∗ q(b−1 ∗ w ∗ b ∗ d) ∗ q(f (b−1 ∗ w ∗ b ∗ d)−1 ) ∗ q(b−1 )] = [q(b) ∗ q(b−1 ) ∗ q(w) ∗ q(b) ∗ q(d) ∗ q((f ◦ d)−1 ) ∗ q((f ◦ b)−1 ) ∗ q((f ◦ w)−1 ) ∗ q(f ◦ b) ∗ q(b−1 )] = [q(b) ∗ q(b−1 ) ∗ q(w) ∗ q(b ∗ d ∗ (f ◦ (b ∗ d))−1 ) ∗ q((f ◦ w)−1 ) ∗ q(f ◦ b) ∗ q(b−1 )] = [q(b) ∗ q(b−1 )] ◦ qπ (γ) ◦ qπ (α) ◦ qπ (fπ (γ)−1 ) ◦ [q(f ◦ b) ∗ q(b−1 )], since γ = [w] and α = [b ∗ d ∗ (f ◦ (b ∗ d))−1 ] = [q(b) ∗ q(b−1 )] ◦ qπ (β) ◦ [q(f ◦ b) ∗ q(b−1 )], since qπ (β) = qπ (γ) ◦ qπ (α) ◦ qπ (fπ (γ)−1 ) = qπ ([b ∗ (b−1 ∗ b ∗ (f ◦ b)−1 ∗ (f ◦ b)) ∗ b−1 ]) = qπ ([b ∗ b−1 ]) = [x¯0 ]. Hence, q ◦ c  q ◦ (f ◦ c).



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If Fq is a q-fixed point class, then from Remark 4.1 it is obvious that  Fq = Fi , i

where the Fi are the ordinary Nielsen fixed point classes. The index of Fq can be calculated by  index (Fq ) = index (Fi ). i

Definition 4.4. Fq is said to be essential if and only if index (Fq , f ) = 0. Definition 4.5. The q-Nielsen number of f , denoted Nq (f ), is the number of essential q-fixed point classes of f . Clearly, Nq (f ) ≤ N(f ). 4.1. The q-Reidemeister number Rq (f ) Definition 4.6. Let q : X → Y be a map from X to Y and let f : X → X be a self-map of X. The q-Reidemeister number of f (denoted Rq (f )) is the number of q-lifting classes. Since the q-Nielsen number of f , Nq (f ), is the number of essential qfixed point classes, clearly Rq (f ) ≥ Nq (f ). 4.2. The algebraic definition of Rq (f ) Definition 4.7. Let α, β ∈ π1 (X, x0 ). We say that β is q-conjugate to α (denoted β ∼q α) if there exists γ ∈ π1 (X, x0 ) such that qπ (β) = qπ (γ) ◦ qπ (α) ◦ qπ (fπ (γ −1 )). If we label a q-fixed point class by [β ◦ f˜]q , then there is a q-conjugacy class of β in π1 (X, x0 ) that corresponds to [β ◦ f˜]q . Thus the number of q-conjugacy classes in π1 (X, x0 ) is equal to the q-Reidemeister number of f , Rq (f ). We refer to the number of q-conjugacy classes as the algebraic definition of Rq (f ).

5. The relation between mod H theory and q-Nielsen theory Theorem 5.1. Let q : X → Y be a map from X to Y and let f : X → X be a self-map of X. Let H = ker(qπ : π1 (X, x0 ) → π1 (Y, q(x0 ))). Elements β and α of the fundamental group π1 (X, x0 ) are mod H f˜π -conjugate if and only if α and β are q-conjugate.

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Proof. Suppose β, α ∈ π1 (X, x0 ) are mod H f˜π -conjugate. This means that there exist h ∈ H = ker(qπ : π1 (X, x0 ) → π1 (Y, q(x0 ))) and γ ∈ π1 (X, x0 ) such that β = h ◦ γ ◦ α ◦ f˜π (γ −1 ). Therefore, qπ (β) = qπ (h ◦ γ ◦ α ◦ f˜π (γ −1 )) = qπ (h)qπ (γ ◦ α ◦ f˜π (γ −1 )) = qπ (γ ◦ α ◦ f˜π (γ −1 )). Thus, α and β are q-conjugate. Conversely, suppose that α and β are q-conjugate. This means that there exists η ∈ π1 (X, x0 ) such that qπ (β) = qπ (η ◦ α ◦ f˜π (η −1 )). Or, qπ (β −1 ◦ η ◦ α ◦ f˜π (η −1 )) = [x0 ]. Therefore, β −1 ◦ η ◦ α ◦ f˜π (η −1 ) ∈ ker(qπ ) = H.



We have shown that the q-Nielsen theory is a special case of mod H theory. In fact, the q-Nielsen theory is the same as the mod H Nielsen theory as we will show in the following subsection. 5.1. Construction of the space Y and the map q for any normal subgroup H of π1 (X, x0 ) Let H be a normal subgroup of the fundamental group π1 (X, x0 ). We will construct a space Y and a map q : X → Y such that H = ker(qπ ).2 Let ai = [αi ] ∈ H be a generator of H. We will construct Y from X by attaching a disc Di to the space X identifying the boundary of Di to the loop αi . Note that Y does not have to be a compact space in our setting in this paper. Therefore, this construction of the space Y is allowed. Hence  Y =X Di ai

and we let the map q : X → Y be the inclusion map. By construction, the kernel of qπ is the subgroup of π1 (X, x0 ) generated by the ai , that is, the group H. This construction, combined with Theorem 5.1, shows that the mod H theory is the same as the q-Nielsen theory. Although q-Nielsen theory is the same as the classical mod H Nielsen theory, the q-Nielsen theory is, in practice, much easier to use than the mod H Nielsen theory as will be illustrated in the following example. Moreover, 2 The

inspiration of this subsection was a conversation with Professor Boju Jiang during the International Conference on Nielsen Theory and Related Topics in June 2009 in Newfoundland, Canada.

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the q-Nielsen theory has already been used in the literature as shown in the following subsection. Example 2. Let X = T ∨ S 1 be the wedge product of a torus and a circle at x0 . The fundamental group of X, then, is π1 (X, x0 ) = a, b, c | aba−1 b−1 . Let f : X → X be a self-map of X such that f induces fπ : π1 (X, x0 ) → π1 (X, x0 ), where fπ (a) = a−1 , fπ (b) = b−1 , fπ (c) = a−1 cb−2 ca. Using the Fadell–Husseini formula in [3], we have RT (f, f˜) = [1] − (−[a−1 ] − [b−1 ] + [a−1 ] + [a−1 cb−2 ]) + [a−1 b−1 ] = [1] + [b−1 ] − [a−1 cb−2 ] + [a−1 b−1 ], where [ · ] denotes the f˜π -conjugacy class. Consider a map q : X → T such that q is an identity of the torus and q sends S 1 to the point x0 . Notice that the image of the normal subgroup H = ker(qπ ) under fπ is not contained in H because fπ (c) is not in H. In this case any α, β ∈ π1 (X, x0 ), where qπ (α) = an1 bm1 , qπ (β) = an2 bm2 , are q-conjugate if and only if n1 ≡ n2 mod 2 and m1 ≡ m2 mod 2. This is due to the fact that for any γ ∈ π1 (X, x0 ), qπ (γ) ◦ qπ (fπ (γ −1 )) = a2s b2t , where s, t are integers. From this, we can easily see that 1, b−1 , a−1 cb−2 and a−1 b−1 are not q-conjugate. Therefore, [1], [b−1 ], [a−1 cb−2 ], [a−1 b−1 ] are distinct f˜π conjugacy classes. Hence we can conclude that N (f ) = 4. 5.2. An estimation of the Nielsen number of a space X = A ∨ B Part of this subsection has already been used in [12] to estimate the Nielsen numbers of maps of wedge products such as a torus wedge surface with boundary, Klein bottle wedge surface with boundary and torus wedge torus. Theorem 5.2 (see [12]). Let X = A ∨ B be a wedge product of two spaces A and B at x0 ∈ X and let f be a self-map on X such that f (A) ⊂ A, then N(q ◦ f |B ) − 1 ≤ N(f ), where q : X → B is the map such that q is the identity on B and q(A) = x0 , and f |B : B → X denotes the restriction of f to B. The proof of this theorem can be found in [12, p. 6]. We will illustrate the use of Theorem 5.2 in a case of nonaspherical wedge spaces which has not been presented in the previous work. Example 3. Let X = T ∨P , where T is a torus and P is a projective plane. Let f : X → X such that f (P ) ⊆ P and the map f induces fπ : π1 (X) → π1 (X)

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where fπ (a) = ca3 b−2 c, fπ (b) = ca4 bc, fπ (c) = c, where a, b are the generators of T and c is the generator of P . If we let q be the map that is the identity on T and maps P to x0 and let f |T denote the restriction of f to T , then the map q ◦ f |T induces (q ◦ f |T )π (a) = a3 b−2 , (q ◦ f |T )π (b) = a4 b. Therefore, N(f ) ≥ N(q ◦ f |T ) − 1 = | det(I − M )| − 1 = |(−2)(0) − (−4)(2)| − 1 = 8 − 1 = 7,   where I is a 2 × 2 identity matrix and M = 34 −2 1 . Acknowledgements This work was started while the author was a graduate student in UCLA under the supervision of Professor Robert F. Brown. It has been many years in the making. The author wishes to thank Professor Brown for his patience, guidance and generosity throughout the entire process of writing this paper, mathematically and otherwise. The author also would like to thank Seungwon Kim for his many helpful suggestions regarding Section 3, while he was a postdoctoral fellow at UCLA [15]. Parts of this work were presented at Nielsen Theory and Related Topics 2009, at Newfoundland, Canada, where Professor Boju Jiang [8] pointed out that the mod H theory is also a special case of the q-Nielsen theory as described in the last part of Section 5. Many thanks to Professor Jiang for that remark. The development of this paper has been supported by the Thailand Research Fund (TRF) MRG5380241 under the mentorship of Professor Robert F. Brown of the Department of Mathematics, UCLA, and Professor Sompong Dhampongsa of the Department of Mathematics, Chiang Mai University. The author wishes to thank both TRF and Professor Dhampongsa as well. Lastly the author would like to thank the referee for all the useful suggestions.

References [1] B. S. Brooks, R. F. Brown, J. Pak and D. H Taylor, Nielsen numbers of maps of tori. Proc. Amer. Math. Soc. 52 (1975), 398–400. [2] R. F. Brown, Fixed point theory. In: History of Topology, North–Holland, Amsterdam, 1999, 271–299. [3] E. Fadell and S. Husseini, The Nielsen number on surfaces. In: Topological Methods in Nonlinear Functional Analysis (Toronto, Ont., 1982), Contemp. Math. 21, Amer. Math. Soc., Providence, RI, 1983, 59–98.

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[4] A. Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002. [5] E. Hart, Algebraic techniques for calculating the Nielsen number on hyperbolic surfaces. In: Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 463–487. [6] E. Hart, The Reidemeister trace and the calculation of the Nielsen number. In: Nielsen Theory and Reidemeister Torsion (Warsaw, 1996), Banach Center Publication 49, Polish Academy of Science, 1999, 151–157. [7] G. Hirsch, D´etermination d’un nombre minimum de points fixes pour certaines repr´esentations. Bull. Sci. Math. 64 (1940), 45–55. [8] B. Jiang, private communication, 2009. [9] B. Jiang, Lectures on Nielsen Fixed Point Theory. Contemp. Math. 14, Amer. Math. Soc., Providence, RI, 1983. [10] M. Kelly, Computing Nielsen numbers of surface homeomorphisms. Topology 35 (1996), 13–25. [11] N. Khamsemanan, An algorithm for calculating the mod H Reidemeister trace of some maps of surfaces with boundary. Ph.D thesis, University of California, Los Angeles, 2006. [12] N. Khamsemanan and S. Kim, Estimating Nielsen number on wedge product spaces. Fixed Point Theory Appl. 2007 (2007), Article ID 83420. [13] N. Khamsemanan, R. Brown, S. Kim, A. Eriksen and K. Merril, Fixed points of maps of a non-aspherical wedge. Fixed Point Theory Appl. 2009 (2009), Article ID 531037. [14] T. Kiang, The Theory of Fixed Point Classes. Springer, Berlin, 1989. [15] S. Kim, private communication, 2005. [16] D. McCord, An estimate of the Nielsen number and an example concering the Lefschetz fixed point theorem. Pacific J. Math. 66 (1976), 195–203. [17] J. Wagner, An algorithm for calculating the Nielsen number on surfaces with boundary. Trans. Amer. Math. Soc. 351 (1999), 41–62. [18] M. Woo and J. Kim, Note on a lower bound of Nielsen number. J. Korean Math. Soc. 29 (1992), 117–125. Nirattaya Khamsemanan Sirindhorn International Institute of Technology (SIIT) Thammasat University P.O. Box 22, Pathum Thani 12121 Thailand e-mail: [email protected]