Nielsen Coincidence Theory Research Articles

Nielsen coincidence theory on infra-solvmanifolds of Sol

We derive averaging formulas for the Lefschetz coincidence numbers, the Nielsen coincidence numbers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type (R). As an application, we compare our formula for the Nielsen coincidence numbers with a result of Jezierski (1992) for pairs of maps on some infra-solvmanifolds of Sol. For all pairs of self-maps of a nonorientable infra-solvmanifold of Sol, we determine the sets of all possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers.

Epsilon Nielsen coincidence theory

Let f,g:X→Y be a pair of maps of a compact, connected, oriented Riemannian manifolds, with or without boundary of the same dimension. In this paper we introduce the concept of the epsilon Nielsen coincidence number Nε(f,g) and develop an algebraic method for computing the epsilon Nielsen number under certain conditions.

Epsilon Nielsen coincidence theory

AbstractWe construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.

Nielsen coincidence theory applied to Borsuk–Ulam geometric problems

This work uses Nielsen coincidence theory to discuss solutions for the geometric Borsuk–Ulam question. It considers triples (X,τ;Y) where X and Y are topological spaces and τ is a free involution on X, (X,τ;Y) satisfies the Borsuk–Ulam theorem if for any continuous map f:X→Y there exists a point x∈X such that f(x)=f(τ(x)). Borsuk–Ulam coincidence classes are defined and a notion of essentiality is defined. The classical Borsuk–Ulam theorem and a version for maps between spheres are proved using this approach.

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension ( k − 1 ) n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.

Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index

Let $M \to B$, $N \to B$ be fibrations and $f_1,f_2\colon M \to N$ be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair $f_1$, $f_2$ over $B$ to a coincidence free pair of maps. In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our $\omega$-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over $B$ are developed. As an illustration we compute e.g the minimal number of coincidence components for all homotopy classes of maps between $S^1$-bundles over $S^1$ as well as their Nielsen and Reidemeister numbers.

On the uniqueness of the coincidence index on orientable differentiable manifolds

The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous mappings on oriented differentiable manifolds, the most common setting for Nielsen coincidence theory.

Nielsen coincidence theory in arbitrary codimensions

Given two maps f1, f2 : Mm → Nn between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC(f1, f2) of pathcomponents of the coincidence space of maps f1′, f2′ where fi′ is homotopic to fi, i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N(f1, f2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N(f1, f2) turns out to coincide with MCC(f1, f2): (i) when m < 2n – 2; (ii) when N is the unit circle; and (iii) when M and N are spheres and a certain injectivity condition involving James-Hopf invariants is satisfied. We also exhibit situations where N(f1, f2) vanishes, but MCC(f1, f2) is strictly positive.

Geometric and homotopy theoretic methods in Nielsen coincidence theory

In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC (and MC , resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are . Furthermore we deduce finiteness conditions for MC . As an application, we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space into path components. Its higher-dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.

Coincidence classes in nonorientable manifolds

We study Nielsen coincidence theory for maps between manifolds of same dimension regardless of orientation. We use the definition of semi-index of a class, review the definition of defective classes, and study the occurrence of defective root classes. We prove a semi-index product formula for lifting maps and give conditions for the defective coincidence classes to be the only essential classes.

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 (i.e. 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 ωe(f), e = + or e = −, 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 link homotopy classification. Our development parallels recent advances in Nielsen coincidence theory and leads also to the notion of Nielsen numbers of link maps. In the special case when N is a product of spheres sample calculations are carried out. They involve the homotopy theory of spheres and, in particular, James–Hopf– invariants.

Obstruction theory and minimal number of coincidences for maps from a complex into a manifold

The Nielsen coincidence theory is well understood for a pair of maps between $n$-dimensional compact manifolds for $n$ greater than or equal to three. We consider coincidence theory of a pair $(f,g)\colon K \to \mathbb N^n$, where $K$ is a finite simplicial complex of the same dimension as the manifold $\mathabb N^n$. We construct an algorithm to find the minimal number of coincidences in the homotopy class of the pair based on the obstruction to deform the pair to coincidence free. Some particular cases are analyzed including the one where the target is simply connected.

REIDEMEISTER CLASSES FOR COINCIDENCE

We generalize algebraic results of Nielsen fixed point theory to Nielsen coincidence theory. We use the algebraic methods of D. Ferrario in Nielsen fixed point theory.

A Van Kampen type theorem for coincidences

The Nielsen coincidence theory is well understood for a pair of maps (f,g) :M n→N n where M and N are compact manifolds of the same dimension greater than two. We consider coincidence theory of a pair (f,g) :K→N n , where the complex K is the union of two compact manifolds of the same dimension as N n . We define a number N( f, g: K 1, K 2) which is a homotopy invariant with respect to the maps. This number is certainly a lower bound for the number of coincidence points, and we prove a minimizing theorem with respect to this number. Finally, we consider the case where the target is a Jiang space and we obtain a nicer description of N( f, g: K 1, K 2) in terms of the Nielsen coincidence numbers of the maps restricted to the subspaces K 1, K 2.

Coincidence theory on the complement

In this work we generalize two aspects of Nielsen fixed point theory on the complement to Nielsen coincidence theory. The first aspect concerns the location (under relative homotopies) of coincidence points. It prepares the way for equivariant coincidence theory and the for second part. A minimum theorem is forthcoming under the condition that the subspace can be by-passed. The second aspect (the study of surplus periodic points on the complement) gives a parallel (but quite different) theory when the subspace cannot be by-passed. Other features of this work include a modified fundamental group approach which simplifies the exposition. Secondly in addition to the usual Jiang condition it includes an analogue of it which ensures that the Reidemeister and Nielsen numbers are the same when the Lefschetz number is nonzero.

On generalizing the Nielsen coincidence theory to non-oriented manifolds

We give an outline of the Nielsen coincidence theory emphasizing differences between the oriented and non-oriented cases.

The Nielsen coincidence theory on topological manifolds

We generalize the coincidence semi-index introduced in [D-J] to pairs of maps between topological manifolds. This permits extending the Nielsen theory to this class of maps.

Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary

Nielsen coincidence theory is extended to manifolds with boundary. For X and Y compact connected oriented n-manifolds with boundary, and for maps ƒ : X → Y and g : ( X, ∂ X) → ( Y, ∂ Y), a coincidence index—which is a local version of Nakaoka's Lefschetz coincidence number—and a Nielsen coincidence number are defined and their properties explored. As an application, coincidence-producing maps g are characterized if Y is acyclic over the rationals and many new examples of coincidence-producing maps are constructed.