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Let X be a compact smooth n-manifold, with or without boundary, and let A be an -dimensional smooth submanifold of the interior of X. Let be a smooth map and be a smooth map whose restriction to A is ϕ. If is an isolated fixed point of f that is a transversal fixed point of ϕ, that is, the linear transformation is nonsingular, then the fixed point index of f at p satisfies the inequality . It follows that if ϕ has k fixed points, all transverse, and the Lefschetz number , then there is at least one fixed point of f in . Examples demonstrate that these results do not hold if the maps are not smooth. MSC:55M20, 54C20.

Fixed points for set-valued mappings from a metric space X (not necessarily complete) intoB(X), the collection of nonempty bounded subsets of X are obtained. The result generalizes some known results.

The first theorem in this paper gives a criterion for detecting a fixed point of a planar homeomorphism into a disc. We then use this result to re-prove the path-connectedness of the space of all planar homeomorphisms fixing only one point, with a given Lefschetz index.

We give some necessary and sufficient conditions for the existence of fixed points of a family of self mappings of a metric space and we establish an equivalent condition for the existence of fixed points of a continuous compact mapping of a metric space.

We develop an intersection theory for definable C p -manifolds in an o-minimal expansion of a real closed field and we prove the invariance of the intersection numbers under definable C p -homotopies ( p>2 ). In particular we define the intersection number of two definable submanifolds of complementary dimensions, the Brouwer degree and the winding numbers. We illustrate the theory by deriving in the o-minimal context the Brouwer fixed point theorem, the Jordan-Brouwer separation theorem and the invariance of the Lefschetz numbers under definable C p -homotopies. A. Pillay has shown that any definable group admits an abstract manifold structure. We apply the intersection theory to definable groups after proving an embedding theorem for abstract definably compact C p -manifolds. In particular using the Lefschetz fixed point theorem we show that the Lefschetz number of the identity map on a definably compact group, which in the classical case coincides with the Euler characteristic, is zero.

We construct noncomplex smooth 4-manifolds which admit genus2 Lefschetz fibrations over S2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures. In this paper we will prove the following theorem. Theorem 1.1. There are infinitely many (pairwise nonhomeomorphic) 4-manifolds which admit genus-2 Lefschetz fibrations but do not carry complex structure with either orientation. Matsumoto [6] showed that S×T #4CP 2 admits a genus-2 Lefschetz fibration over S with global monodromy (β1, ..., β4), where β1, ..., β4 are the curves indicated by Figure 1. (For definitions and details regarding Lefschetz fibrations see [6], [5].)

The purpose of this paper is to obtain a generalization of the famous Browder's fixed point theorem and some equivalent forms. As application, these results are utilized to study the existence problems of fixed points and nearest points.

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.

A new simplicial homtopy algorithm is presented for computing the Leray-Schauder fixed points as well as Merrill fixed points and Eaves fixed points. Moreover, a coercivity condition to guarantee the computation proceeding in a bounded region is given.

We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdnK →K is expanding directions preserving.

We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

We apply the Lefschetz Fixed Point Theorem to show that every square matrix over the quaternions hasright eigenvalues. We classify them and discuss some of their properties such as an analogue of Jordan canonical form and diagonalization of elements of the compact symplectic group Sp(n).

We present the geometric method for detecting periodic solutions of time periodic nonautonomous differential equations in interior of convex subset of euclidean space. The method is based on the Lefschetz fixed point theorem and the topological principle of Wazewski. Two applications to the existence of positive periodic solutions are considered.

Two Picard numbers and two Lefschetz numbers are defined for a real algebraic surface. They are similar to the Picard number and the Lefschetz number of a complex algebraic surface. For these numbers, some estimates and relations in the form of inequalities are proved.

New fixed-point theorems and random fixed-point theorems are presented for multivalued maps.

A selector theorem for non-convex orientor fields on closed manifolds is given and the Lefschetz fixed point theorem is used to establish an existence result for these ones.

Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p * : H n ( G ) → H n ( M ) is nonzero, where n = dim M . In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M . Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f , which can be computed algebraically.

Let X be a metrizable space and let (C, E) be a pair of compact ANRs contained in X. For a given continuous map f : C → X, we call (C, E) proper with respect to f if C ∩ f(E) ⊂ E and C ∩ f(C)⧹C ⊂ E . For such a pair we introduce the so-called transfer endomorphism f#(C,E) of H ∗ (C, E) . The first main theorem of this note asserts that if the Lefschetz number Λ(f#(C,E)) is nonzero then f has a fixed point in C⧹E . In order to state the second main result we assume that X is an ENR and there exists a finite family (C1, E1),…, (Cn, En) of proper ENR-pairs with respect to f, Ci ⊂ Ei − 1, such that f has no fixed points in the boundary of C1, in En, and in the complements of the relative interiors of Ci in Ei − 1. The second theorem asserts that under these hypotheses the fixed point index ind (f, int C1) is equal to ∑ni = 1 Λ(f#(Ci,Ei)). We provide an example indicating how to apply that theorem in order to determine the fixed point index of a Poincare map of a nonautonomous ordinary differential equations, and how to use that index to a result on bifurcation of periodic solutions.

Under consideration are some nonclassical analogues of the Cartan fixed point theorem which are stated for meromorphic self-maps of complex manifold with nontrivial Bergman form. We discuss their exactness and examine possibilities for fixed point conditions on these manifolds.

We generalize the Lefschetz coincidence theorem to non-oriented manifolds. We use (co-) homology groups with local coefficients. This generalization requires the assumption that one of the considered maps is orientation true.

WOS: 000326916300013, In this article we study the fixed point properties of digital images. Moreover, we prove the Lefschetz fixed point theorem for a digital image. We then give some examples about the fixed point property. We conclude that sphere-like digital images have the fixed point property.

For a compact Lie group G, we use G-equivariant Poincar\'e duality for ordinary RO(G)-graded homology to define an equivariant intersection product, the dual of the equivariant cup product. Using this, we give a homological construction of the equivariant Lefschetz number and a simple proof of the equivariant Lefschetz fixed point theorem. With similar techniques, an equivariant Fuller index with values in the rationalized Burnside ring is constructed., Comment: 33 pages

We prove a Lefschetz formula for graph endomorphisms , where G is a general finite simple graph and ℱ is the set of simplices fixed by T. The degree of T at the simplex x is defined as , a graded sign of the permutation of T restricted to the simplex. The Lefschetz number is defined similarly as in the continuum as , where is the map induced on the k th cohomology group of G. The theorem can be seen as a generalization of the Nowakowski-Rival fixed-edge theorem (Nowakowski and Rival in J. Graph Theory 3:339-350, 1979). A special case is the identity map T, where the formula reduces to the Euler-Poincare formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed-point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. If is the automorphism group of a graph, we look at the average Lefschetz number . We prove that this is the Euler characteristic of the chain and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function is a product of two dynamical zeta functions and, therefore, has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits. MSC:58J20, 47H10, 37C25, 05C80, 05C82, 05C10, 90B15, 57M15, 55M20.

In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin(I), etc) are needed at a few points in Chap. 5.

We study the Hilbert function and the graded Betti numbers for “generic” linear quotients of Artinian standard graded algebras, especially in the case of Weak Lefschetz algebras. Moreover, we investigate a particular property of Weak Lefschetz algebras, the Betti Weak Lefschetz Property, which makes possible to completely determine the graded Betti numbers of a generic linear quotient of such algebras.

In this chapter we discuss topics of invariant theory such as coinvariant algebras of reflection groups. In particular the coinvariant algebras of real reflection groups have the SLP, and the set of Lefschetz elements is explicitly determined in most cases.

We use the geometric data to define a bordism invariant for the fiberwise intersection theory. Under some certain conditions, this invariant is an obstruction for the theory. Moreover, we prove the converse of fiberwise Lefschetz fixed point theorem.

has been considered by Srzednicki [11 13], using a combination of generalized Conley's isolating blocks, the index of singular points of autonomous systems and the Lefschetz fixed point theorem. Recently, it has been shown that slight extensions of Srzednicki's results could be obtained in a simpler and more elementary way using continuation theorems of the Leray Schauder type (see [6, 9]). The aim of the present paper is to show that the same methodology can be adapted to prove the existence of periodic solutions for more general classes of equations and systems. article no. 0054

We give a new proof of the following result due to Duan: Let f : X → X f\,:\, X \to X be any self-map of a finite connected H H -space. Then f k f^{k} has a fixed point if k ≥ 2 k \geq 2 . Our proof is based on an approach due to Steve Halperin, whereby the Lefschetz number of a self-map is expressed in terms of the eigenvalues of the induced homomorphism of rational homotopy groups. This allows us to give a considerably shorter proof which avoids most of the technicalities of the original proof.

Let X X be a closed surface, and let f : X → X f: X \rightarrow X be a map. We would like to determine Min ( f ) := m i n { | F i x g | : g ≃ f } . \text {Min}(f):= \mathrm {min} \{ | \mathrm {Fix}g| : g \simeq f\}. Nielsen fixed point theory provides a lower bound N ( f ) N(f) for Min ( f ) \text {Min}(f) , called the Nielsen number, which is easy to define geometrically and is difficult to compute. We improve upon an algebraic method of calculating N ( f ) N(f) developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle K K . Nielsen numbers for maps on K K were first calculated (geometrically) by Halpern. We include a sketch of Halpern’s never published proof that N ( f ) = Min ( f ) N(f)= \text {Min}(f) for all maps f f on K K .

We present two theorems on periodic solutions of nth-order ordinary differential equations with the right-hand side T-periodic with respect to the time variable. We apply a topological method based on the Lefschetz Fixed Point Theorem and the Wazewski Principle.

We study the set of periodic points and the set of periods of different classes of [Formula: see text] self maps of a compact manifold. We give sufficient conditions in order that these sets be infinite. Our main tools are the Lefschetz fixed point theory and homology.

Let X be a connected, locally finite polyhedron. For U a compact, connected subpolyhedron of X, let f: U → X be a map with Fix(f) ∩ ∂U = O. The local H-Nielsen number N(f) is a lower bound for the number of fixed points of any map in a restricted homotopy class of f. We prove that the local generalized H-Lefschetz number L H (f; f , \ gi) can, in certain cases, be expressed in terms of the isotropy subgroups of the local Reidemeister action and the local Lefschetz numbers of the lifts of f. From L H (f; f , \ gi) we determine N(f). We consider applications to lens spaces and to a spherical space form.

The existence of essential fixed points is proved for compact self-maps of arbitrary absolute neighborhood retracts, provided the generalized Lefschetz number is nontrivial and the topological dimension of a fixed point set is equal to zero. Furthermore, continuous self-maps of some special compact absolute neighborhood retracts, whose Lefschetz number is nontrivial, are shown to possess pseudo-essential fixed points even without the zero dimensionality assumption. Both results are applied to the existence of essential and pseudo-essential multivalued fractals. An illustrative example of this application is supplied.

Let X be a uniquely arcwise-connected continuum and f : X → X f:X \to X a map which is arc-preserving (i.e., the image of each arc is an arc or a point). We prove that f has a fixed point.

The Lefschetz number and fixed point index can be thought of as two different descriptions of the same invariant. The Lefschetz number is algebraic and defined using homology. The index is defined more directly from the topology and is a stable homotopy class. Both the Lefschetz number and index admit generalizations to coincidences and the comparison of these invariants retains its central role. In this paper we show that the identification of the Lefschetz number and index using formal properties of the symmetric monoidal trace extends to coincidence invariants. This perspective on the coincidence index and Lefschetz number also suggests difficulties for generalizations to a coincidence Reidemeister trace., Minor revision

It has been remarked by Hartshorne, that Barth’s theorem for a smooth projective X follows from the strong Lefschetz theorem for the cohomology of X. Using the strong Lefschetz theorem for intersection cohomology, we give an extension of Barth’s theorem to singular X. This naturally raises several questions concerning possible Barth theorems on the level of intersection cohomology.

‎In this paper‎, ‎we define a new cone ball-metric and get fixed points and common fixed points for the Meir-Keeler type functions in cone ball-metric spaces‎.