Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

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.     

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.     

Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree

We construct natural self-maps of compact cohomogeneity one manifolds and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields relations between the order of the Weyl group and the Euler characteristic of a principal orbit. As examples we determine all cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type that lead to self-maps of degree ≠ −1; 0; 1. We derive explicit formulas for new coordinate polynomial self-maps of the compact matrix groups SU(3), SU(4), and SO(2n). For SU(3) we determine precisely which integers can be realized as degrees of self-maps. Supported by a DFG Heisenberg scholarship and DFG priority program SPP 1154.     

Holomorphic Lefschetz formula for manifolds with boundary

The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f:MM in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschetz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschetz formula on a strictly convex domain in ⁿ, n>1.Mathematics Subject Classification (2000):32S50; 58J20*Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.**Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.     

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

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.     

A Runge Approximation Theorem for Pseudo-Holomorphic Maps

The Runge approximation theorem for holomorphic maps is a fundamental result in complex analysis, and, consequently, many works have been devoted to extend it to other spaces (e.g. maps between certain algebraic varieties or complex manifolds). This article presents such a result for pseudo-holomorphic maps from a compact Riemann surface to a compact almost-complex manifold M, given that the manifold M admits many pseudo-holomorphic maps from ${\mathbb {C}{\rm P}^1}$ which can be thought of as local approximations of the Laurent expansion az +br ²/z. This result specializes to some compact algebraic varieties (e.g. rationally connected projective varieties). An application to Lefschetz fibrations is presented.     

Formality of Donaldson submanifolds

We introduce the concept of s–formal minimal model as an extension of formality. We prove that any orientable compact manifold M, of dimension 2n or (2n−1), is formal if and only if M is (n−1)–formal. The formality and the hard Lefschetz property are studied for the Donaldson submanifolds of symplectic manifolds constructed in [13]. This study permits us to show an example of a Donaldson symplectic submanifold of dimension eight which is formal simply connected and does not satisfy the hard Lefschetz theorem. An erratum to this article is available at .     

The degree of multivalued maps from manifolds to spheres

In this paper, for closed connected oriented manifolds M and N of the same dimension, we study the degree of a triple (??, p, q), where p is a Vietoris map from a compact space ?? to M and q is a continuous map from ?? to N. In particular, we have Borsuk?CUlam-type degree theorems on manifolds with involutions.     

Automorphism invariant Cartan subgroups and power maps of disconnected groups

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.     

Fixed point theorems in fuzzy metric spaces

In the present work, we establish several fixed point theorems for a new class of self-maps in M-complete fuzzy metric spaces and compact fuzzy metric spaces, respectively.     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

Weak shadowing for discrete dynamical systems on nonsmooth manifolds

In J. Math. Anal. Appl. 189 (1995) 409-423, Corless and Pilyugin proved that weak shadowing is a C⁰ generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set.     

A Note on the Set of Periods of Transversal Homological Sphere Self-maps

Let M be a n -dimensional manifold with the same homology than the n -dimensional sphere. A C 1 map f : M M M is called transversal if for all m ] N the graph of f m intersects transversally the diagonal of M 2 M at each point ( x , x ) such that x is a fixed point of f m . We study the minimal set of periods of f by using the Lefschetz numbers for periodic points. In the particular case that n is even, we also study the set of periods for the transversal holomorphic self-maps of M .     

Periodic manifolds with spectral gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number N we construct periodic manifolds such that the essential spectrum of the corresponding Laplacian has at least N open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

Families of simply connected 4-manifolds with the same Seiberg-Witten invariants

This article presents several new constructions of infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic. While the construction gives strong evidence that any two of these manifolds of are not diffeomorphic, they cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, diffeomorphic seems to be a very difficult question to answer. For one of these constructions, each member of the family is symplectic with the further property that each contains nullhomologous tori with the property that infinitely many log transformations on these tori yield nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin-Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on ellipitic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer g?3, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.