Nielsen Number Research Articles

A Nielsen theory for coincidences of iterates

As the title suggests, this paper gives a Nielsen theory of coincidences of iterates of two self maps f,g:X→X of a closed manifold X. The idea is, as much as possible, to generalize Nielsen type periodic point theory, but there are many obstacles. Familiar results as in periodic point theory are obtained, but often require stronger hypotheses.

The Wecken property for random maps on surfaces with boundary

A selfmap is Wecken when the minimal number of fixed points among all maps in its homotopy class is equal to the Nielsen number, a homotopy invariant lower bound on the number of fixed points. All selfmaps are Wecken for manifolds of dimension not equal to 2, but some non-Wecken maps exist on surfaces.We attempt to measure how common the Wecken property is on surfaces with boundary by estimating the proportion of maps which are Wecken, measured by asymptotic density. Intuitively, this is the probability that a randomly chosen homotopy class of maps consists of Wecken maps. We show that this density is nonzero for surfaces with boundary.When the fundamental group of our space is free of rank n, we give nonzero lower bounds for the density of Wecken maps in terms of n, and compute the (nonzero) limit of these bounds as n goes to infinity.

Nielsen numbers of selfmaps of Sol 3-manifolds

We compute the Nielsen number N(f) of a selfmap f of a closed three-dimensional manifold with Sol-geometry. In particular, we show that for a self homeomorphism f, the Nielsen number N(f) is either 0 or 4 and the Reidemeister number R(f) is either ∞ or 4, respectively.

The asymptotic density of Wecken maps on surfaces with boundary

The Nielsen number $N(f)$ is a lower bound for the minimal number of fixed points among maps homotopic to $f$. When these numbers are equal, the map is called Wecken. A recent paper by Brimley, Griisser, Miller, and the second author investigates the abundance of Wecken maps on surfaces with boundary, and shows that the set of Wecken maps has nonzero asymptotic density. We extend the previous results as follows: When the fundamental group is free with rank $n$, we give a lower bound on the density of the Wecken maps which depends on $n$. This lower bound improves on the bounds given in the previous paper, and approaches 1 as $n$ increases. Thus the proportion of Wecken maps approaches 1 for large $n$. In this sense (for large $n$) the known examples of non-Wecken maps represent exceptional, rather than typical, behavior for maps on surfaces with boundary.

Dynamics of random selfmaps of surfaces with boundary

We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps.In this sense, we show that ``almost all'' such selfmaps have periodic points of every period, and that in fact the number of periodic points of period $n$ grows exponentially in $n$. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps.  &nbsp Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.

The mod H Nielsen Theory and the q-Nielsen Theory

Let \({f: X \rightarrow X}\) be a self-map of a topological space. In 1992, M. Woo and J. Kim introduced the q-Nielsen theory which is a tool to estimate the Nielsen number. In this paper, we compare the q-Nielsen theory to the mod H Nielsen theory. We develop an algebraic formulation of the mod H theory which does not require the condition \({f(H) \subset H}\) based on the universal covering space instead of the regular cover, as well as an algebraic formulation of the q-Nielsen theory. In fact, we show that the q-Nielsen theory is the same as the mod H theory for \({H = {\rm ker}(q_\pi : \pi_1(X) \rightarrow \pi_1(Y ))}\) , where qπ is the induced homomorphism of q and Y is a topological space. We also provide applications of the q-Nielsen theory and the mod H theory in the case where f(H) is not contained in H, as well as applications for wedge product spaces of nonaspherical types.

The WYK algorithm for maps of aspherical figure-eight type finite polyhedra

Let X be an aspherical figure-eight type finite polyhedron. In [16], Yi and Kim proved that there is an algorithm for computing the Nielsen number of Yi-type self-maps of X. In this paper, we complete the algorithm for computing the Nielsen number of arbitrary self-maps of X.

Minimum numbers and Wecken theorems in topological coincidence theory. I

Minimum numbers measure the obstruction to removing coincidences of two given maps (between smooth manifolds M and N of dimensions m and n, resp.). In this paper, we compare them to four distinct types of Nielsen numbers. These agree with the classical Nielsen number when m = n (e.g., in the fixed point setting where M = N and one of the maps is the identity map). However, in higher codimensions, m − n > 0, their definitions and computations involve distinct aspects of differential topology and homotopy theory. We develop tools which help us (1) to decide when a minimum number is equal to a Nielsen number (“Wecken theorem”), and (2) to determine Nielsen numbers. Here certain homotopy theoretical criteria play a central role. E.g., failures of the “Wecken condition” (cf. Definition 1.18 below) can have very interesting geometric consequences. The selfcoincidence case where the two maps are homotopic turns out to be particularly illuminating. We give many concrete applications in special settings where M or N are spheres, spherical space forms, projective spaces, tori, Stiefel manifolds or Grassmannians. Already in the simplest examples an important role is played, e.g., by Kervaire invariants, all versions of Hopf invariants (à la James, Hilton, Ganea, . . . ) and the elements in the stable homotopy of spheres defined by invariantly framed Lie groups.

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.

FIXED POINTS AND COINCIDENCES IN TORUS BUNDLES

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper, we investigate fiberwise analoga and present a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They also allow a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S1)2-bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.

Local Fixed Point Indices of Iterations of Planar Maps

Let f : U → R 2 be a continuous map, where U is an open subset of R 2 . We consider a fixed point p of f which is neither a sink nor a source and such that { p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations {ind( f n , p)} ∞=1 is periodic, bounded from above by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem (Annals of Math., 146, 241-293 (1997)) onto the class of non-injective maps. We apply ourresult to study the dynamics of continuous maps on 2-dimensional sphere.

Equivariant Nielsen fixed point theory for n-valued maps

We develop an equivariant Nielsen fixed point theory for n-valued G-maps by associating (as in Better (2010) [2]) an abstract simplicial complex to any equivariant n-valued map and defining, in terms of this complex, two n-valued continuous G-homotopy invariants that are lower bounds for the number of fixed points and of orbits in the n-valued continuous G-homotopy class of a given n-valued G-map. We also provide an equivariant Hopf construction for n-valued G-maps as well as a minimality result for the Nielsen numbers introduced in this setting.

Reidemeister coincidence invariants of fiberwise maps

Given two fiberwise maps f 1 , f 2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of ( f 1 , f 2 ) as the orbit set of a group action of π 1 ( B ) . The size and number of orbits captures crucial extra information. E.g. for torus bundles of arbitrary dimensions over the circle this determines the minimum coincidence numbers of the pair ( f 1 , f 2 ) completely. In particular we can decide when f 1 and f 2 can be deformed away from one another or when a fiberwise self-map can be made fixed point free by a suitable homotopy. In two concrete examples we calculate the minimum and Nielsen numbers for all pairs of fiberwise maps explicitly. Odd order orbits turn out to play a special rôle.

Nielsen Type Numbers of Self-Maps on the Real Projective Plane

Employing the induced endomorphism of the fundamental group and using the homotopy classification of self-maps of real projective plane , we compute completely two Nielsen type numbers, (f) and (f), which estimate the number of periodic points of f and the number of fixed points of the iterates of map f.

Poly(vinyl chloride)/metallic oxides/organically modified montmorillonite nanocomposites: Preparation, morphological characterization, and modeling of the mechanical properties

AbstractPoly(vinyl chloride), metallic oxides (from copper, molybdenum, and zinc), and organically modified montmorillonite (O‐MMT) nanocomposites were prepared in a melt‐blending or intercalation‐in‐the‐molten‐state process. The morphology of the nanocomposites was evaluated with X‐ray diffraction (XRD) and transmission electron microscopy (TEM). Properties, such as the mechanical, thermal, and electrical properties, and the dynamic thermal stability against dehydrochlorination were also evaluated. Nanocomposites with a hybrid intercalated/exfoliated structure were obtained in all of the situations considered, as demonstrated by the XRD and TEM results and indirectly by the increment of Young's modulus of the formulations with increasing amount of O‐MMT incorporated. The modeling of Young's modulus by the Halpin–Tsai, Hui–Shia, and Lewis–Nielsen theories showed that the process of nanocomposite preparation allowed the aspect ratio of the clay particles to increase; these values were comparable to those nanocomposites obtained by other researchers with different polymeric matrices and methodologies. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

Coincidence Nielsen numbers for covering maps for smooth manifolds

Let f , g : M → N be maps between closed smooth manifolds of the same dimension, and let p : M ˜ → M and p ′ : N ˜ → N be finite regular covering maps. If the manifolds are nonorientable, using semi-index, we introduce two new Nielsen numbers. The first one is the Linear Nielsen number N L ( f , g ) , which is a linear combination of the Nielsen numbers of the lifts of f and g. The second one is the Nonlinear Nielsen number N ED ( f , g ) . It is the number of certain essential classes whose inverse images by p are inessential Nielsen classes. In fact, N ( f , g ) = N L ( f , g ) + N ED ( f , g ) , where by abuse of notation, N ( f , g ) denotes the coincidence Nielsen number defined using semi-index.

Fixed Points of Maps of a Nonaspherical Wedge

Let be a finite polyhedron that has the homotopy type of the wedge of the projective plane and the circle. With the aid of techniques from combinatorial group theory, we obtain formulas for the Nielsen numbers of the selfmaps of .

Averaging Formula for Nielsen Numbers of Maps on Infra-Solvmanifolds of Type (R)

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group Ψ and let f: M → M be a continuous map. The averaging formula for Nielsen numbersis proved. This is a workable formula for the difficult number N(f).

Nielsen numbers of n-valued fiber maps

The Nielsen number for n-valued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of n-valued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of n-valued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of single-valued maps. If the fibration is orientable, the product formula for single-valued fiber maps fails to generalize, but a “semi-product formula" is obtained. In this way, the class of n-valued multimaps for which the Nielsen number can be computed is substantially enlarged.

COMPUTATION OF THE NIELSEN TYPE NUMBERS FOR MAPS ON THE KLEIN BOTTLE

Let f : M <TEX>${\rightarrow}$</TEX> M be a self-map on the Klein bottle M. We compute the Lefschetz number and the Nielsen number of f by using the infra-nilmanifold structure of the Klein bottle and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer n, we provide an explicit algorithm for a complete computation of the Nielsen type numbers <TEX>$NP_n(f)$</TEX> and <TEX>$N{\Phi}_{n}(f)\;of\;f^{n}$</TEX>.