Given two maps f_1, f_2 : M^m \longrightarrow N^n between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC (f_1, f_2) of pathcomponents of the coincidence space of maps f'_1, f'_2 where f'_i is homotopic to f_i, i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N (f_1, f_2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N (f_1, f_2) turns out to coincide with MCC (f_1, f_2): (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 (f_1, f_2) vanishes, but MCC (f_1, f_2) is strictly positive.
In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1, f_2). Furthermore we deduce finiteness conditions for MC(f_1, f_2). As an application we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decom…
In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs (f_1,f_2) of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC(f_1,f_2) (and MC(f_1,f_2), respectively) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1,f_2). Furthermore, we deduce finiteness conditions for MC(f_1,f_2). As an application we compute both minimum numbers explicitly in v…
Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility properties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems to be important for a more comprehensive understanding.
As the title suggests, this paper gives a Nielsen theory of coincidences of iterates of two self maps f, g of a closed manifold. The ideas is, as much as possible, to generalize Nielsen type periodic point theory, but there are many obstacles. Many times we get similar results to the "classical ones" in Nielsen periodic point theory, but with stronger hypotheses.
Coincidences of maps between smooth manifolds are studied via a geometric approach which involves (nonstabilized) normal bordism theory and pathspaces.
Let $f_1,...,f_k:M\to N$ be maps between closed manifolds, $N(f_1,...,f_k)$ and $R(f_1,...,f_k)$ be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we relate $R(f_1,...,f_k)$ with $R(f_1,f_2),...,R(f_1,f_k)$. When $N$ is a torus or a nilmanifold, we compute $R(f_1,...,f_k)$ which, in these cases, is equal to $N(f_1,...,f_k)$.
Suppose X,Y are manifolds, f,g:X->Y are maps. The well-known Coincidence Problem studies the coincidence set C={x:f(x)=g(x)}. The number m=dimX-dimY is called the codimension of the problem. More general is the Preimage Problem. For a map f:X->Z and a submanifold Y of Z, it studies the preimage set C={x:f(x) in Y}, and the codimension is m=dimX+dimY-dimZ. In case of codimension 0, the classical Nielsen number N(f,Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take i…
Let M to B, N to B be fibrations and f1,f2 :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 f1,f2 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 ω-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 compo…
We discuss coincidences of pairs (f_1, f_2) of maps between manifolds. We recall briefly the definition of four types of Nielsen numbers which arise naturally from the geometry of generic coincidences. They are lower bounds for the minimum numbers MCC and MC which measure to some extend the 'essential' size of a coincidence phenomenon. In the setting of fixed point theory these Nielsen numbers all coincide with the classical notion but in general they are distinct invariants. We illustrate this by many examples involving maps from spheres to the real, complex or quaternionic projective space K…
Given two maps between smooth manifolds, the obstruction to removing their coincidences (via homotopies) is measured by minimum numbers. In order to determine them we introduce and study an infinite hierarchy of Nielsen numbers N_i, i = 0, 1, ..., \infty. They approximate the minimum numbers from below with decreasing accuracy, but they are (in principle) more easily computable as i grows. If the domain and the target manifold have the same dimension (e.g. in the fixed point setting) all these Nielsen numbers agree with the classical definition. However, in general they can be quite distinct. …
Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand the resulting bordism class of coincidence data and the corresponding Nielsen numbers are strong looseness obstructions. On the other hand the values which these invariants may possibly assume turn out to satisfy severe restrictions, e.g. the Nielsen numbers can only take the values 0, 1 or the cardinality of the fundamental group of N. In order to show…
Let $ f_1, f_2 \colon X^m \longrightarrow Y^n $ be maps between smooth connected manifolds of the indicated dimensions $ \!m\! $ and $ \!n \!\!\!$. Can $ f_1, f_2 $ be deformed by homotopies until they are coincidence free (i.e. $ f_1(x) \neq f_2(x) $ for all $ x \in X $)? The main tool for addressing such a problem is tradionally the (primary) Nielsen number $ N(f_1, f_2) $. E.g. when $ m < 2n - 2 $ the question above has a positive answer precisely if $ N(f_1, f_2) = 0 $. However, when $ m = 2n - 2 $ this can be dramatically wrong, e.g. in the fixed point case when $ m = n = 2 $. Also, in a …
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.
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 N…
Given a link map f into a manifold of the form Q = N \times \Bbb R, when can it be deformed to an unlinked position (in some sense, e.g. where its components map to disjoint \Bbb R-levels) ? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions \widetildeω_ε(f), 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 link homotopy classification. Our development parallels recent adva…
When can a map between manifolds be deformed away from itself? We describe a (normal bordism) obstruction which is often computable and in general much stronger than the classical primary obstruction in cohomology. In particular, it answers our question completely in a large dimension range. As an illustration we give explicit criteria in three sample settings: projections from Stiefel manifolds to Grassmannians, sphere bundle projections and maps defined on spheres. In the first example a theorem of Becker and Schultz concerning the framed bordism class of a compact Lie group plays a central …
This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations…
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 Nielse…
Given two fiberwise maps f1, f2 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 (f1,f2) as the orbit set of a group operation of π1(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 (f1,f2) completely. In particular we can decide when f1 and f2 can be deformed away from one another or when a fiberwise selfmap can be ma…
In this article we studied Nielsen coincidence theory for maps between manifolds of same dimension without hypotheses on orientation. We use the definition of semi-index of a class, we review the definition of defective classes and study the appearance of defective root classes. We proof a semi-index product formula type for lifting maps and we presented conditions such that defective coincidence classes are the only essencial classes.
In this paper we continue to study (`strong') Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with covering maps, paying special attention to selfcoincidence questions. As a sample application we calculate each of these numbers for all maps from spheres to (real, complex, or quaternionic) projective spaces. Our results turn out to be intimately related to recent work of D Gonca…
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 represent 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) dimensio…
The aim of this paper is to generalize some of the properties and results regarding both the coincidence point set and the common fixed point set of any two digitally continuous maps to the case of several (more than two) digitally continuous mappings. Moreover, we study how rigidity may affect these coincidence and homotopy coincidence point sets. Also, we investigate whether an established result by Staecker in Nielsen classical topology regarding the coincidence set for many maps still remains valid in the digital topological setting.
Let $f,g:X\to Y$ be maps from a compact infra-nilmanifold $X$ to a compact nilmanifold $Y$ with $\dim X\ge \dim Y$. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number $N(f,g)$ vanishes then $f$ and $g$ are deformable to be coincidence free. We also show that if $X$ is a connected finite complex $X$ and the Reidemeister coincidence number $R(f,g)=\infty$ then $f\sim f'$ so that $C(f',g)=\{x\in X \mid f'(x)=g(x)\}$ is empty.
In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for t…
Let $f,g: X\to G/K$ be maps from a closed connected orientable manifold $X$ to an orientable coset space $M=G/K$ where $G$ is a compact connected Lie group, $K$ a closed subgroup and $\dim X=\dim M$. In this paper, we show that if $L(f,g)=0$ then $N(f,g)=0$; if $L(f,g)\ne 0$ then $N(f,g)=R(f,g)$ where $L(f,g), N(f,g)$, and $R(f,g)$ denote the Lefschetz, Nielsen, and Reidemeister coincidence numbers of $f$ and $g$, respectively. When $\dim X> \dim M$, we give conditions under which $N(f,g)=0$ implies $f$ and $g$ are deformable to be coincidence free.
