Reducing the number of fixed points of some homeomorphisms on nonprime 3-manifolds.
Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers
Let f be a smooth self-map of an m-dimensional (m ≥ 4) closed connected and simply-connected manifold such that the sequence of the Lefschetz numbers 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 defined in combinatorial terms and is constant for all sufficiently large r. We compute J[f] for self-maps...
Reidemeister conjugacy for finitely generated free fundamental groups
Let X be a space with the homotopy type of a bouquet of k circles, and let f: X → X be a map. In certain cases, algebraic techniques can be used to calculate N(f), the Nielsen number of f, which is a homotopy invariant lower bound on the number of fixed points for maps homotopic to f. Given two fixed points of f, x and y, and their corresponding group elements, and , the fixed points are Nielsen equivalent if and only if there is a solution z ∈ π₁(X) to the equation . The Nielsen number is the...
Reidemeister orbit sets
The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending Ferrario's work on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets. Similar formulae for Nielsen type essential orbit numbers are also proved for fibre preserving maps.
Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action
Let (X,A) be a pair of topological spaces, T : X → X a free involution and A a T-invariant subset of X. In this context, a question that naturally arises is whether or not all continuous maps have a T-coincidence point, that is, a point x ∈ X with f(x) = f(T(x)). In this paper, we obtain results of this nature under cohomological conditions on the spaces A and X.
Remark on ANR-divisors
Remarks on minimal round functions
We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions.
Remarks on the n-dimensional geometric measure of compacta
Repulsive Fixed Points of Multivalued Transformations and the Fixed Point Index.
Residual sets not of maximum dimension
Rétractes Absolus de Voisinage Algébriques
2000 Mathematics Subject Classification: 54C55, 54H25, 55M20.We introduce the class of algebraic ANRs. It is defined by replacing continuous maps by chain mappings in Lefschetz’s characterization of ANRs. To a large extent, the theory of algebraic ANRs parallels the classical theory of ANRs. Every ANR is an algebraic ANR, but the class of algebraic ANRs is much larger; the most striking difference between these classes is that every locally equiconnected metrisable space is an algebraic ANR, whereas...
Revisiting Cauty's proof of the Schauder conjecture.
Rigidity Properties of Compact Lie Groups Modulo Maximal Tori.
Roots of mappings from manifolds.
Sandwich type theorems
Seiberg-Witten invariants, the topological degree and wall crossing formula
Following S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b 1 = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.
Selections and near-selections in metric linear spaces without local convexity
We characterize the AR property in convex subsets of metric linear spaces in terms of certain near-selections.
Sequences of fixed point indices of iterations in dimension 2.
Smith equivalence and finite Oliver groups with Laitinen number 0 or 1.
Smith theory and quasi-periodicity in Bredon cohomology