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Nielsen coincidence theory in arbitrary codimensions
 J. reine angew. Math
"... Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M  f1(x) = f2(x)} as small (or simple in some sense) as possi ..."
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Cited by 22 (5 self)
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Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M  f1(x) = f2(x)} as small (or simple in some sense) as possible after possibly deforming f1 and f2 by a homotopy. Question. How large is the minimum number of coincidence components MCC(f1,f2): = min{#π0(C(f ′ 1,f ′ 2))  f ′ 1 ∼ f1,f ′ 2 ∼ f2}? In particular, when does this number vanish, i.e. when can f1 and f2 be deformed away from one another? This is a very natural generalization of one of the central problems of classical fixed point theory (where M = N and f2 = identity map): determine the minimum number of fixed points among all maps in a given homotopy class (see [Br] and [BGZ], proposition 1.5). Note, however, that in higher codimensions m − n> 0 the coincidence locus is generically a closed (m−n)manifold so that it makes more sense to count pathcomponents rather than points. Also the methods of (first order, singular) (co)homology will no longer be strong enough to capture the subtle geometry of coincidence manifolds. In this lecture I will use the language of normal bordism theory (and a nonstabilized version thereof) to define and study lower bounds N(f1,f2) (and N #(f1,f2)) for MCC(f1,f2). After performing an approximation we may assume that the map (f1,f2) : M → N × N is smooth and transverse to the diagonal ∆ = {(y,y) ∈ N × N  y ∈ N}. Then the coincidence locus C = C(f1,f2) = (f1,f2) −1 (∆) is a closed smooth (m − n)dimensional manifold, equipped with i) maps
The Lefschetz coincidence theory for maps between spaces of different dimensions
, 2000
"... For a given pair of maps f, g: X → M from an arbitrary topological space to an nmanifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg: H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X ..."
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Cited by 8 (5 self)
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For a given pair of maps f, g: X → M from an arbitrary topological space to an nmanifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg: H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X such that f(x) = g(x).
The three faces of Nielsen: coincidences, intersections and preimages
"... In addition to the many different Nielsentype numbers that have been introduced to study fixed points, there are Nielsentype numbers that have been created to study coincidences of maps; intersections of maps; and preimages of maps. These problems, while distinct, are all similar, and the Niels ..."
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Cited by 6 (0 self)
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In addition to the many different Nielsentype numbers that have been introduced to study fixed points, there are Nielsentype numbers that have been created to study coincidences of maps; intersections of maps; and preimages of maps. These problems, while distinct, are all similar, and the Nielsen theories that have been created to study them display strong structural similarities as well. In this paper, we explore these similarities, and show that the relations between the three theories are closer and more formal than just similarity. There are transformations that allow any of the three Nielsen problems to be converted into either of the other two. Analysis of these transformations allows us to make precise the relationships between the three Nielsen theories. 1 Introduction A prominent feature of Nielsen theory has been the development of "Nielsentype" theories. Nielsen theory was originally developed to study fixed points of selfmaps f : X ! X of compact polyhedra. B...
Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms
"... In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs.f1;f2 / of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbe ..."
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Cited by 6 (5 self)
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In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs.f1;f2 / 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.f1;f2 / (and MC.f1;f2/, resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to.f1;f2/. Furthermore, we deduce finiteness conditions for MC.f1;f2/. As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E.f1;f2 / into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf–Ganea homomorphisms which turn out to yield finiteness obstructions for MC. 55M20, 55Q25, 55S35, 57R90; 55N22, 55P35, 55Q40 1
Geometric and homotopy theoretic methods in Nielsen coincidence theory
, 2006
"... In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f1, f2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f1, f2) (and MC(f1, f2), resp.) of pathcompone ..."
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Cited by 5 (4 self)
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In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f1, f2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f1, f2) (and MC(f1, f2), resp.) of pathcomponents (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f1, f2). Furthermore we deduce finiteness conditions for MC(f1, f2). 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 E(f1, f2) into pathcomponents. 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.
Estimating Nielsen Numbers On Infrasolvmanifolds
, 1992
"... . A wellknown lower bound for the number of fixed points of a selfmap f : X \Gamma! X is the Nielsen number N(f ). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number L(f ), on the other hand, is readily computable, but does not give a lower bound for the number of f ..."
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Cited by 4 (1 self)
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. A wellknown lower bound for the number of fixed points of a selfmap f : X \Gamma! X is the Nielsen number N(f ). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number L(f ), on the other hand, is readily computable, but does not give a lower bound for the number of fixed points. In this paper, we investigate conditions on the space X which guarantee either N(f) = jL(f)j or N(f) jL(f)j. By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) jL(f)j for all selfmaps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable group of finite index). Moreover, for infranilmanifolds, there is a Lefschetz number formula which computes N(f ). 1. Estimating Nielsen Numbers Consider a continuous self map f : X \Gamma! X. Let F ix(f) denote the fixed point set fx 2 X j f(x) = xg. One of the fundamental problems of fixed point theory is to estimate (preferably from below) the cardinality of this se...
A Nielsen theory for intersection numbers
 Fund. Math
, 1997
"... Nielsen theory, originally developed as a homotopytheoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. In this paper, the techniques of Nielsen theory are applied to the study of int ..."
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Cited by 3 (1 self)
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Nielsen theory, originally developed as a homotopytheoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. In this paper, the techniques of Nielsen theory are applied to the study of intersections of maps. A Nielsentype number, the Nielsen intersection number NI(f; g), is introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, NI(f; g) gives a lower bound for the number of points of intersection for all maps homotopic to f and g. 1 Introduction Nielsen fixed point theory, a homotopytheoretic approach to fixedpoint theory, grew out of Nielsen's work in the 1920's on surface homeomorphisms. From those origins, Nielsen fixed point theory has grown into a richly developed theory for fixed points. Moreover, the methods of Nielsen theory have been translated from fixed point problems into other domains, such ...
REMOVING COINCIDENCES OF MAPS BETWEEN MANIFOLDS OF DIFFERENT DIMENSIONS
, 2003
"... We consider sufficient conditions of local removability of coincidences of maps f, g: N → M, where M, N are manifolds with dimensions dim N ≥ dim M. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. ..."
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Cited by 3 (3 self)
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We consider sufficient conditions of local removability of coincidences of maps f, g: N → M, where M, N are manifolds with dimensions dim N ≥ dim M. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. We also address the normalization property of the index and coincidenceproducing maps.
Reidemeister coincidence invariants of fiberwise maps
"... 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 pi1(B). The size and number of o ..."
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Cited by 1 (0 self)
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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 pi1(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 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 role.
A Nielsen theory for coincidences of iterates
, 2012
"... 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 ..."
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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. Kewords: Nielsen numbers; coincidences; iterates; periodic points; roots; manifolds