On Lefschetz periodic point free self-maps

We study the periodic point free maps and Lefschetz periodic point free maps on connected retract of a finite simplicial complex using the Lefschetz numbers. We put special emphasis in the self-maps on the product of spheres and of the wedge sums of spheres.


Introduction and statement of the main results
Let X be a topological space and f a continuous self-map on X. We say that a point x in X is a periodic point of f of period m if f m (x) = x and f j (x) = x for 1 ≤ j ≤ m − 1; if m = 1, x is called a fixed point. We say that f is a periodic point free map, if it does not have periodic points.
Let X be a retract of a finite simplicial complex, see [7] for a precise definition. The compact manifolds, the CW complexes are spaces of this type. Let n be the topological dimension of X. If f : X → X is a continuous map on X, it induces a homomorphism on the kth rational homology group of X for 0 ≤ k ≤ n, i.e., f * k : H k (X, Q) → H k (X, Q). The H k (X, Q) is a finite dimensional vector space over Q and f * k is a linear map whose matrix has integer entries, then Lefschetz number of f is defined as The Lefschetz fixed point theorem states that if L(f ) = 0 then f has a fixed point, see for more details [3,11].
The Lefschetz fixed point theorem and its improvements is one of the most used tools for studying the existence of fixed points and periodic points, for continuous self-maps on compact manifolds, among other references see for instance [1][2][3][4]8,13].
The map f is called Lefschetz periodic point free if L(f m ) = 0, for m ≥ 1. Clearly the periodic point free maps are Lefschetz periodic point free.
Note that if f is periodic point free, then f is Lefschetz periodic point free, but in general the converse does not hold. Periodic point free and Lefschetz periodic point free maps have been studied previously, see for instance [5,6,12,13].
In the present note, we give necessary and sufficient conditions for a continuous self-map in a connected retract of a finite simplicial complex to be a Lefschetz periodic free (Theorem 1 and Corollary 2). These conditions are given in term of the eigenvalues of the induced maps on homology. In Theorems 3 and 4, we give criteria for a map on the product of spheres to be periodic point free and Lefschetz periodic point free. In Theorem 5, we give a criteria for a map on the wedge sums of spheres to be periodic point free and Lefschetz periodic point free.
Let Λ k be the set of eigenvalues of f * k , and Λ := ∪ n k=1 Λ k , i.e., the set of all eigenvalues of the induced maps on the homology by f .
Let λ ∈ Λ, we define where mult k (λ) is the multiplicity of λ as eigenvalue of f * k , if λ is not an eigenvalue of f * k , then mult k (λ) = 0. The quantity e(λ) counts the multiplicities of λ as eigenvalue of all maps f * k with k even, minus the multiplicities of λ as eigenvalue of all maps f * k with k odd.  We consider the case of periodic point and Lefschetz periodic point free maps on the product of spheres. It is well known that if X is the n-dimensional torus and f : X → X is a continuous map, then f is Lefschetz periodic point free if and only if 1 is an eigenvalue of f * 1 , see for instance [1,8,14]. A similar result is known for maps on the product of spheres of the same dimension. Let X = S n × · · · × S n and f : X → X is a continuous map. If n is odd then f is Lefschetz periodic point free if and only if 1 is an eigenvalue of f * 1 . If n is even then f is never periodic point free, see [13].
Vol. 20 (2018) On Lefschetz periodic point free self-maps Page 3 of 9 38 Theorem 3. Let X = S n1 ×· · ·×S n l and let f : X → X be a continuous map. If the numbers n i are even for all 1 ≤ i ≤ l, then f is not Lefschetz periodic point free.
The combinatorics of the periodic structure of self-maps on X could be very complicated in general. In the following particular case, we give a criteria when a self-map on a product of spheres of different dimensions are Lefschetz free periodic. Let We suppose that n 1 < · · · < n l . By elementary combinatorics we have that the cardinality of the set is at most 2 l −1. We shall assume that the cardinality of M is exactly 2 l − 1, i.e., the numbers n 1 , . . . , n l are such that all the sums defined in the set M are different.
Theorem 4. Let X = S n1 × · · · × S n l with n 1 < · · · < n l and let f : X → X be a continuous map. Assume that the cardinality of M is 2 l − 1. Then the following statements hold. Theorems 3 and 4 are proved in Sect. 3. Given topological spaces X and Y with chosen points x 0 ∈ X and y 0 ∈ Y , then the wedge sum X ∨ Y is the quotient of the disjoint union X and Y obtained by identifying x 0 and y 0 to a single point (see for more details pp. 10 of [7]). The wedge sum is also known as "one point union". For example, S 1 ∨ S 1 is homeomorphic to the figure "8," two circles touching at a point. We can think the wedge sums of spheres as generalization of graphs in higher dimensions. In the following theorem, we consider conditions for a continuous self-map on the wedge sum of spheres to be periodic point free.
(a) If 0 is not an eigenvalue of f * k for 1 ≤ k ≤ n l and − 1 + (−1) n1+1 s 1 + · · · + (−1) n l +1 s l = 0, then f is not periodic point free. Theorem 5 is proved in Sect. 4. If l = 1, and n 1 = 1, then X is the circle, a particular case of a graph. Periodic point free maps on graphs were studied in [12].
If l = 1 and n 1 > 1, then X is the n 1 -dimensional sphere, and a continuous self-map f of X is not periodic point free, unless n 1 is odd, 1 is a simple eigenvalue of f * n1 , and 0 is eigenvalue of f * n1 with multiplicity s 1 − 1. This fact follows from expression of the Lefschetz zeta function of f which in this case is ζ f (t) = (1 − t) −1 q 1 (t) (−1) n 1 +1 .

Proof of Theorem 1
Let X be a connected retract of a finite simplicial complex, and f be a continuous self-map on X. Then the Lefschetz zeta function of f is defined as This function keeps the information of the Lefschetz number for all the iterates of f , so this function gives information about the set of periods of f .
Note that if f is Lefschetz periodic point free if and only if ζ f (t) = 1.
There is an alternative way to compute the Lefschetz zeta function, namely where n = dim X, n k = dim H k (X, Q), Id := Id * k is the identity map on H k (X, Q), and by convention det(Id * k − tf * k ) = 1 if n k = 0, for a proof see [4].

Lemma 6. The Lefschetz zeta function of X can be expressed in the form
Proof. We consider This completes the proof of the lemma. From Lemma 6 it follows that the zero eigenvalues of f * k for 1 ≤ k ≤ n do not contribute to any non-trivial factor in the expression (3) of the Lefschetz zeta function.
Since X is connected then 1 is a simple eigenvalue of f * 0 , then the term (1 − t) −1 is a factor of (3). Hence, 1 is an eigenvalue of f * k , for some k = 0, and it is required that k≥1 (−1) k mult k (1) = −1 to have e(1) = 0.
From the previous arguments it is clear that if conditions (a) and (b) of Theorem 1 hold, then f is Lefschetz periodic point free. This completes the proof of the theorem.

Proofs of Theorems 3 and 4
Let X = S n1 × · · · × S n l . Using the Künneth Theorem (see for instance [7]), we compute the homology groups of X over the rational numbers Q. They are given by with 0 ≤ k ≤ n 1 + · · · + n k ; where b k is the number of ways that k can be obtained as by summing up subsets of (n 1 , . . . , n l ), i.e., b k is the cardinality of the set ⎧ ⎨ The numbers b k are called the Betti numbers of X.
Proof of Theorem 3. We remark that b k = 0 if and only if k ∈ M . So the formula (2) of the Lefschetz zeta function can be expressed in the following manner: where q k (t) = det(Id * k − tf * k ). Since the numbers n i are even, clearly all the elements of the set M are even numbers. Therefore, cannot be equal 1, hence the map f is not Lefschetz periodic point free.
Let X = S n1 × · · · × S n l with n 1 < · · · < n l and we assume that the cardinality of M is 2 l − 1. In this case, the Betti numbers of X are b k = 1 if k ∈ M and b k = 0, otherwise, i.e., and trivial otherwise.
If f : X → X is a continuous map, the formula (2) can be written as where f * k = (a k ).
Proof of statement (a) of Theorem 4 The sum of the degree of the polynomials of numerator of (4) minus the degree of the polynomials in the denominator of (4) is So, if this sum is not zero then ζ f (t) = 1. Hence, f is not periodic point free.
Proof of statement (b) of Theorem 4 If f has the property described, then its Lefschetz zeta function is Since for each i ∈ M , with i = j and i ≡ 1 (mod 2), there exists i satisfying i ≡ 0 (mod 2), and a i = a i , we have ζ f (t) = 1.
In this case, the possible situations are: (1) If n 1 , n 2 and n 3 are even, then f is not periodic point free. As it is a particular case of Theorem 3. (2) If n 1 , n 2 and n 3 are odd, then (3) In the case that two n i are odd and the other even. We can suppose that n 1 and n 2 are odd and n 3 even. Then Vol. 20 (2018) On Lefschetz periodic point free self-maps Page 7 of 9 38 (4) In the case that two n i are even and the other odd. We can suppose that n 1 and n 2 are even and n 3 odd. Then In the cases (2), (3) and (4) the map f is not periodic point free, unless the condition of statement (b) of Theorem 4 is satisfied.
In the following lines, we consider for a product of three spheres a situation when n 1 , n 2 , n 3 do not satisfy that the cardinal of the set M is 2 3 −1 = 7, but it satisfies that this cardinal is 6, being n 3 = n 1 + n 2 . By the Künneth formula we have that the homology groups of X are if k = n 1 , n 2 , n 1 + n 3 , n 2 + n 3 , n 1 + n 2 + n 3 ; otherwise. Hence, where q n3 (t) = det(Id * n3 − tf * n3 ). The degree of the polynomial q n3 (t) is at most 2, and it is equal to 2 if and only if 0 is not an eigenvalue of f n3 . In this case, the possible situations are: (1) If n 1 and n 2 are even, then the numerator of the Lefschetz zeta function is 1 and the denominator is a polynomial of degree greater than 1. Hence, ζ f (t) = 1. Therefore, f it is not periodic point free. This is a particular case of Theorem 3. (2) If n 1 and n 2 are odd. Then (3) If n 1 and n 2 have different parities, say n 1 even and n 2 odd, then a n1+n2+n3 t)(1 − a n1 t)(1 − a n2+n3 t) .
In the cases (2) and (3), ζ f (t) = 1 if the eigenvalues of the induced maps on homology are different from 0 and 1. However, in (2) and (3) it is possible that ζ f (t) = 1 and consequently the map f be Lefschetz periodic point free.
As we can see the combinatorial analysis of the Lefschetz zeta function becomes quite complicated when the numbers n 1 , . . . , n l do not satisfy that the cardinal of the set M is 2 l − 1.