Lefschetz fixed point theorem for intersection homology.
Leray endomorphisms and cone maps
Linking and coincidence invariants
Given a link map f into a manifold of the form Q = N × ℝ, when can it be deformed to an “unlinked” position (in some sense, e.g. where its components map to disjoint ℝ-levels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions , ε = + or ε = -, which often answer this question completely and which, in addition, turn out to distinguish a great number of different link homotopy classes. In certain cases they even allow a complete...
Local constribution to the Lefschetz fixes point formula.
Locally admissible multi-valued maps
In this paper we generalize the class of admissible mappings as due by L. Górniewicz in 1976. Namely we define the notion of locally admissible mappings. Some properties and applications to the fixed point problem are presented.
Mapping theorems for compacta with an arbitrary involution
Maps into the torus and minimal coincidence sets for homotopies
Let X,Y be manifolds of the same dimension. Given continuous mappings , i = 0,1, we consider the 1-parameter coincidence problem of finding homotopies , 0 ≤ t ≤ 1, such that the number of coincidence points for the pair is independent of t. When Y is the torus and f₀,g₀ are coincidence free we produce coincidence free pairs f₁,g₁ such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the Lefschetz...
Merging of degree and index theory.
Méthodes homologiques dans la théorie des applications et des champs de vecteurs sphériques dans les espaces de Banach [Book]
Minimal component numbers of fixed point sets
Let f: (X,A) → (X,A) be a relative map of a pair of compact polyhedra. We introduce a new relative homotopy invariant , which is a lower bound for the component numbers of fixed point sets of the self-maps in the relative homotopy class of f. Some properties of are given, which are very similar to those of the relative Nielsen number N(f;X,A).
Minimal fixed point sets of relative maps
Let f: (X,A) → (X,A) be a self map of a pair of compact polyhedra. It is known that f has at least N(f;X,A) fixed points on X. We give a sufficient and necessary condition for a finite set P (|P| = N(f;X,A)) to be the fixed point set of a map in the relative homotopy class of the given map f. As an application, a new lower bound for the number of fixed points of f on Cl(X-A) is given.
Minimal Nielsen root classes and roots of liftings.
Minimal number of periodic points for smooth self-maps of S³
Let f be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension m ≥ 3 and r a fixed natural number. A topological invariant , introduced by the authors [Forum Math. 21 (2009)], is equal to the minimal number of r-periodic points for all smooth maps homotopic to f. In this paper we calculate for all self-maps of S³.
Minimal periods of maps of rational exterior spaces
The problem of description of the set Per(f) of all minimal periods of a self-map f:X → X is studied. If X is a rational exterior space (e.g. a compact Lie group) then there exists a description of the set of minimal periods analogous to that for a torus map given by Jiang and Llibre. Our approach is based on the Haibao formula for the Lefschetz number of a self-map of a rational exterior space.
Minimal surface maps, fixed point indices and a Jiang-Guo type inequality
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...
Multi-valued maps of subsets of Euclidean spaces
Necklace bisection with one cut less than needed.
Nielsen fixed point theory and symplectic Floer homology
We describe a connection between Nielsen fixed point theory and symplectic Floer homology for surfaces. A new asymptotic invariant of symplectic origin is defined.
Nielsen fixed point theory on manifolds
The study of fixed points of continuous self-maps of compact manifolds involves geometric topology in a significant way in topological fixed point theory. This survey will discuss some of the questions that have arisen in this study and indicate our present state of knowledge, and ignorance, of the answers to them. We will limit ourselves to the statement of facts, without any indication of proof. Thus the reader will have to consult the references to find out how geometric topology has contributed...