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Agraïments: The first author was partially supported by Fundación Séneca de la Región de Murcia, grant number 08667/PI/08 and Junta de Comunidades de Castilla-La Mancha, grant number PEII09-0220-0222. We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: CPn the n-dimensional complex projective space, HPn the n-dimensional quaternion projective space, Sn the n-dimensional sphere and Sp ×Sq the product space of the p-dimensional with the q-dimensional spheres.

In this paper, we study common fixed point properties for a family of mappings. We present the extensions of Caristi's fixed point theorem and Markov–Kakutani fixed point theorem for a family of set-valued mappings.

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S4n−1 → S2n ×S m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.

Song derives certain multi-variable rational identities by studying torus actions on some homogeneous manifolds and applying the Atiyah–Bott–Segal–Singer Lefschetz fixed point theorem. In this paper, we give a direct proof of these rational identities by using the q-Lucas theorem. Moreover, we also give a similar new rational identity.

In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite T 0 {T_{0}} -spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite T 0 {T_{0}} -spaces. However, using the notion of the subdivision of a finite T 0 {T_{0}} -space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle 𝕊 1 {\mathbb{S}^{1}} .

Recently, Wardowski (Fixed Point Theory Appl. 2012:94, 2012) introduced a new concept of F-contraction and proved a fixed point theorem which generalizes the Banach contraction principle. Following this direction of research, in this paper, we present some new fixed point results for F-expanding mappings, especially on a complete G-metric space.

New B-U-cyclic weak contraction C-class function concept has been introduced, B-U-cyclic weak F-φ-Φ-contraction types of mapping are defined and the existence of fixed point for such types has been proved. These results mainly generalize fixed point theorems in some previous research papers.

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n -dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n -dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7] . The methods used are based on the arithmetical properties of the number n .

A fixed point index theory is developed for a class of nowhere normal-outward compact maps defined on a cone which do not necessarily take values in the cone. This class depends on the retractions on the cone and contains self-maps for any retractions, and weakly inward maps and generalized inward maps when the retraction is a continuous metric projection. The new index coincides with the previous fixed point index theories for compact self-maps and generalized inward compact maps. New fixed point theorems are obtained for nowhere normal-outward compact maps and applied to treat some abstract boundary value problems and Sturm-Liouville boundary value problems with nonlinearities changing signs.

In this paper, a common fixed point theorem for contractive type fuzzy mappings in a complete metric space is proved due to Cho (2005) [1]. Further an example is given for the results of Cho (2005) [1, Theorem 3.1] and Park and Jeong (1997) [2, Theorem 3.2] which are not satisfying the condition “for all x , y ∈ X ” and have a fixed point.

In this paper, we define the notion of graph trace kernels as a generalization of trace kernels. We associate a microlocal Lefschetz class with a graph trace kernel and prove that this class is functorial with respect to the composition of kernels. We apply graph trace kernels to the microlocal Lefschetz fixed point formula for constructible sheaves., Comment: 18 pages, revised, to appear in Publ. RIMS

This article provides a survey concerning Lefschetz fixed point formula and Lefschetz coincidence formula in the smooth and singular cases, moreover we show a Lefschetz type formula for the Coincidence number of two maps. As a consequence we obtain a relation with correspondences, and we provide some examples.

In this paper, as a survey paper, we review many works related to fixed point theory for digital spaces using Lefschetz fixed point theorem, Banach fixed point theorem, Nielsen fixed point theorem and so forth. Besides, we refer some properties of the fixed point property of a digital k-retract.

In Huang and Lee (in The Refined Analytic Torsion and a Well-posed Boundary Condition for the Odd Signature Operator, 2010) introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of global boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map which is a local isometry on the boundary and has only simple fixed points. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

Let A be a Poincare duality algebra over the field F and G a finite group of grading preserving algebra automorphisms of A . The subject of this manuscript is the fixed subalgebra A G . We show that in the nonmodular case, if G fixes a fundamental class of A then A G is again a Poincare duality algebra, and if A has the strong Lefschetz property with a strong Lefschetz element L ∈ A 1 left fixed by G then A G also has the strong Lefschetz property.

For a compact symplectic manifold which is s-Lefschetz which is weaker than the hard Lefschetz property, we prove that the Lefschetz decomposition for de Rham cohomology also holds.

In this paper, we prove a fixed point theorem for a self map in fuzzy metric space where the map satisfies a different condition.

We consider a condition under which the projectivization P (Ek) of a complex k-bundle Ek → M over an even-dimensional manifold M can have the hard Lefschetz property, affected by [4]. It depends strongly on the rank k of the bundle Ek. Our approach is purely algebraic by using rational Sullivan minimal models [2]. We will give some examples. The contents of this paper is according to [6]. A Poincare duality space Y of the formal dimension fd(Y ) = max{i;H (Y ;Q) = 0} = 2m is said to be cohomologically symplectic (c-symplectic) if u = 0 for some u ∈ H(Y ;Q) and, furthermore, is said to have the hard Lefschetz property (or simply the Lefschetz property) with respect to the c-symplectic class u, if the maps ∪u : Hm−j(Y ;Q) → H(Y ;Q) 0 ≤ j ≤ m are monomorphisms (then called the Lefschetz maps) [7]. (Then we often say that H∗(Y ;Q) is a Lefschetz algebra [4].) For example, a compact Kahler manifold has the hard Lefschetz property [7], [3, Theorem 4.35]. Let M be an even-dimensional manifold and ξ : E → M be a complex k-bundle over M . The projectivization of the bundle ξ P (ξ) : CP k−1 j → P (E) → M satisfies the rational cohomology algebra condition (∗) : H∗(P (E);Q) = H∗(M ;Q)[x] (xk + c1xk−1 + · · ·+ cjxk−j + · · ·+ ck−1x+ ck) where ci are the Chern classes of ξ and x is a degree 2 class generating the cohomology of the complex projective space fiber (Leray-Hirsch theorem) [1], [4], [7, p.122]. The manifold P (E) appears as the exceptional divisor in blow-up construction for a certain embedding of M [5], [7, Chap.4]. When M is a non-toral symplectic nilmanifold of dimension 2n, there is a bundle E such that P (E) is not Lefschetz [8], [4, Example 4.4]. In general, for a 2k-dimensional manifold M and a fibration

In this paper, we establish a common fixed point theorem in semi-metric space with compatible mapping of type (E) which improves and extends similar known results in the literature.

In this paper we prove a common fixed point theorem of eight self mappings satisfying a generalized inequality using the concept of compatibility.

In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.

On a subset of Banach space which is a locally finite union of closed, convex sets, we extend the Lefschetz fixed point theorem for set-valued mappings. As an application of this result we give a partial answer to Nussbaum’s conjecture for set-valued mappings.

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the th cyclotomic polynomial, when is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

Let X be an arbitrary topological space and let Y be a closed connected oriented n-dimensional manifold. In this work we consider p maps f 1,...,fp : X → Y, p ≥ 2, define a Lefschetz class L(f 1,...,fp ) ∈ H n(p-1)(X;ℚ) and prove that L(f 1,...,fp ) ≠ 0 implies f 1(x) = f 2(x) = ⋯ = fp (x) for some x ∈ X. In the particular case where Y is a homology sphere there are presented some formulas to calculate L(f 1,...,fp ).

We give a necessary and sufficient condition in terms of a matrix for which all Tate classes are Lefschetz for simple abelian varieties over an algebraic closure of a finite field. As an application, we prove under an assumption that all Tate classes are Lefschetz for simple factors of the Jacobian variety of a Fermat curve of prime degree.

We introduce the notion of generalized -admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

Let f be a smooth self-map of an m-dimensional (m >3) closed connected and simply-connected manifold such that the sequence of the Lefschetz num- bers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defned in combinatorial terms and is constant for all suciently large r. We compute J[f] for self-maps of some manifolds with simple structure of homology groups.

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda$ of the map $f_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ even. We prove that if $f$ is $C^1$ having finitely many periodic points all of them hyperbolic, then $f$ is almost quasi-unipotent.

Let f be a self-map of a compact connected manifold M . We characterize Lefschetz periodic point free continuous self-maps of M for several classes of manifolds and generalize the results of Guirao and Llibre [J.L.G. Guirao, J. Llibre, On the Lefschetz periodic point free continuous self-maps on connected compact manifolds, Topology Appl. 158 (16) (2011) 2165–2169].

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations., 17 pages, 18 figures

We construct a spectral sequence converging to symplectic homology of a Lefschetz fibration whose E1 page is related to Floer homology of the monodromy symplectomorphism and its iterates. We use this to show the existence of fixed points of certain symplectomorphisms.

In this work, we introduce a modified notion of weakly F -contractive mappings and prove new coupled fixed point theorems for the mappings which are weakly F -contractive. Also, we give an example to validate the main results in this work.

Let R = ⊕i=0∞Ri be a connected graded commutative algebra over the field ℚ of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f) on the ℚ vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f). The main tool we used is the graded algebra version of Grobner basis theory. At the end of this paper, some examples and applications are given.

In this paper we prove that a deformed tensor product of two Lefschetz algebras is a Lefschetz algebra. We then use this result in conjunction with some basic Schubert calculus to prove that the coinvariant ring of a finite reflection has the strong Lefschetz property., 31 pages

Summary. In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its n + 1 anely indepedent vertices of E n . First we introduce the Lebesgue number, which for an arbitrary open cover of a compact metric space M is a positive real number so that any ball of about such radius must be completely contained in a member of the cover. Then we introduce the notion of a bounded simplicial complex and the diameter of a bounded simplicial complex. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of E n . This article is based on [16].

S. Hu and Y. Sun[1] defined the fixed point index for weakly inward mappings, investigated its properties and studied fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we further investigate boundary conditions, under which the fixed point index for i(A, Ω, p) is equal to nonzero, where i(A, Ω, p) is the completely continuous and weakly inward mapping. Correspondingly, we can obtain many new fixed point theorems of the completely continuous and weakly inward mapping, which generalize some famous theorems such as Rothe’s theorem, Altman’s theorem, Petryshyn’s theorem etc. in the case of weakly inward mappings. In addition, our conclusions extend the famous fixed point theorem of cone expansion and compression to the case of weakly inward mappings. Moreover, the main results contain and generalize the corresponding results in the recent work[2].

We present several new Lefschetz fixed point results for compact self maps in new classes of spaces with respect to the map.

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. We associate with such a current an equivariant cyclic cohomology class of Connes' C*-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points. As special cases, we recover a number of classical results, and since we may use any closed Haefliger current on the foliation, we get new and very interesting formulae., This is an updated version of the paper submitted in November 2009. It will appear in the Journal of Functional Analysis.

In the present paper, we prove some Krasnosel’skii–Leray–Schauder type fixed point theorems for weak topology. Some fixed point theorems for the sum of two weakly sequentially continuous mappings are also presented. Our results extend and improve on ones from several earlier works.