Transversality and Lefschetz numbers for foliation maps

Let be a smooth foliation on a closed Riemannian manifold M, and let Λ be a transverse invariant measure of . Suppose that Λ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the Λ-Lefschetz number of any leaf preserving diffeomorphism is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological Λ-Lefschetz number is equal to the analytic Λ-Lefschetz number defined by Heitsch and Lazarov, which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to . Dedicated to V. I. Arnold     

Lefschetz numbers for maps of fiber spaces

Expressions for the Lefschetz numbers of maps of fiber spaces in terms of those of maps of the fibers and the base are obtained. A similar problem is considered for Lefschetz coincidence numbers.     

The Lefschetz coincidence theorem in o-minimal expansions of fields

In this paper we prove the Lefschetz coincidence theorem in o-minimal expansions of fields using the o-minimal singular homology and cohomology.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Proximal maps,prox maps and coincidence points

Given a continuous seminorm p on a Hausdorff locally convex space X and nonempty subsets A, B of X, let Proximal(B;A) (resp. Prox(A,B)) denote the set of p-proximal points of B in A (resp. the set of ordered pairs of p-proximal points of the pair (A,B)). In this article we identify suitable families A,B of nonempty subsets of X and natural topologies on them with a view to study continuity properties of B →Proximal( B;A) and (A,B) →Prox(A,B). This leads us to obtain a best approximation result and a coincidence theorem for multifunctions defined on non-compact sets.     

Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.     

Essential maps and coincidence theory

Using an elementary approach, we present new coincidence principles for general classes of maps.     

Asymptotic Lefschetz fixed point theory for ANES(compact) maps

A number of new Lefschetz fixed point theorems are established for ANES(compact) maps. Also compact absorbing contraction maps are discussed.      

Hochschild Lefschetz class for \mathcal{D }-modules

We introduce a notion of Hochschild Lefschetz class for a good coherent $\mathcal{D }$ -module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann–Roch formula for a $\mathcal{D }$ -module on a compact complex orbifold.     

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky-MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.     

Selection principles for maps of several variables

We show that a pointwise precompact sequence of maps from the n-dimensional rectangle into a metric semigroup, whose total variations in the sense of Vitali, Hardy and Krause are uniformly bounded, contains a pointwise convergent subsequence. We present a variant of this result for maps with values in a reflexive separable Banach space with respect to the weak pointwise convergence of maps.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

Deterministic and random coincidence point results for f-nonexpansive maps

Some deterministic and random coincidence theorems for f-nonexpansive maps are obtained. As applications, invariant approximation theorems are derived. Our results unify, extend and complement various known results existing in the literature.     

Locally covering maps in metric spaces and coincidence points

We study the notion of α-covering map with respect to certain subsets in metric spaces. Generalizing results from [1] we use this notion to give some coincidence theorems for pairs of single-valued and multivalued maps one of which is relatively α-covering while the other satisfies the Lipschitz condition. These assertions extend some classical contraction map principles. We define the notion of α-covering multimap at a point and give conditions under which the covering property of a multimap at each interior point of a set implies that it is covering on the whole set. As applications we consider the solvability of a system of inclusions and the existence of a positive trajectory for a semilinear feedback control system. This paper is dedicated to Professor Felix Browder on the occasion of his jubilee     

Lefschetz decomposition for de Rham cohomology on weakly Lefschetz symplectic manifolds

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.