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17
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).
NIELSEN ROOT THEORY AND HOPF DEGREE THEORY
 PACIFIC JOURNAL OF MATHEMATICS VOL. 198, NO. 1
, 2001
"... ... a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f −1 (c). There is a formula for calculating N(f; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orie ..."
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Cited by 7 (0 self)
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... a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f −1 (c). There is a formula for calculating N(f; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orientable, and also to manifolds that have nonempty boundaries and are not compact, provided that the map f is boundarypreserving and proper. Because of its connection with degree theory, we introduce the transverse Nielsen root number for maps transverse to c, obtain computational results for it in the same setting, and prove that the two Nielsen root numbers are sharp lower bounds in dimensions other than 2. We apply these extended root theory results to the degree theory for maps of not necessarily orientable manifolds introduced by Hopf in 1930. Thus we reestablish, in a new and modern treatment,
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
A Lefschetztype coincidence theorem

, 1999
"... A Lefschetztype coincidence theorem for two maps f, g: X → Y from an arbitrary topological space to a manifold is given: I fg = λ fg, that is, the coincidence index is equal to the Lefschetz number. It follows that if λ fg � = 0 then there is an x ∈ X such that f(x) = g(x). In particular, the theo ..."
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Cited by 5 (3 self)
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A Lefschetztype coincidence theorem for two maps f, g: X → Y from an arbitrary topological space to a manifold is given: I fg = λ fg, that is, the coincidence index is equal to the Lefschetz number. It follows that if λ fg � = 0 then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains wellknown coincidence results for (i) X, Y manifolds, f boundarypreserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “pointlike” (acyclic) and “spherelike” values.
Minimizing Coincidence Numbers of Maps into Projective Spaces
, 2008
"... 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 ..."
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Cited by 5 (4 self)
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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. Gonçalves and D. Randall concerning maps which can be deformed away from themselves but not by small deformations; in particular, there are close connections to the
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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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.
Higher order Nielsen numbers
"... Abstract. Suppose X, Y are manifolds, f, g: X → Y are maps. The wellknown Coincidence Problem studies the coincidence set C = {x: f(x) = g(x)}. The number m = dim X − dim Y 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, i ..."
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Abstract. Suppose X, Y are manifolds, f, g: X → Y are maps. The wellknown Coincidence Problem studies the coincidence set C = {x: f(x) = g(x)}. The number m = dim X − dim Y 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) ∈ Y}, and the codimension is m = dim X + dim Y − dim Z. 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 into account other topological characteristics of C. The goal is to find a “lower estimate ” of the bordism group Ωp(C) of C. The answer is the Nielsen group Sp(f, Y) defined as follows. In the classical definition the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let S ′ p(f, Y) = Ωp(C) / ∼N, then the Nielsen group of order p is the part of S ′ p(f, Y) preserved under homotopies of f. The Nielsen number Np(F, Y) of order p is the rank of this group (then N(f, Y) = N0(f, Y)). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided. 1.
Selfcoincidences and roots in Nielsen theory
"... Abstract. Given two maps f1 and f2 from the sphere S m to an nmanifold 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 one hand the resulting bordism class of coincidence data ..."
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Abstract. Given two maps f1 and f2 from the sphere S m to an nmanifold 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 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 this we compare different Nielsen classes in the root case (where f1 or f2 is constant) and we use the fact that all but possibly one Nielsen class are inessential in the selfcoincidence case (where f1 = f2). Also we deduce strong vanishing results.
The relative coincidence Nielsen number
"... by Jerzy J e z i e r s k i (Warszawa) Abstract. We define a relative coincidence Nielsen number Nrel(f, g) for pairs of maps between manifolds, prove a Wecken type theorem for this invariant and give some formulae expressing Nrel(f, g) by the ordinary Nielsen numbers. Introduction. In [S2] pairs of ..."
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by Jerzy J e z i e r s k i (Warszawa) Abstract. We define a relative coincidence Nielsen number Nrel(f, g) for pairs of maps between manifolds, prove a Wecken type theorem for this invariant and give some formulae expressing Nrel(f, g) by the ordinary Nielsen numbers. Introduction. In [S2] pairs of spaces A ⊂ X and maps f: X → X such that f(A) ⊂ A were considered. A relative Nielsen number of such maps was defined, i.e. a lower bound of the cardinality of fixed points which is invariant with the respect to homotopies preserving A. In this paper we generalize this construction to coincidences. We consider pairs of maps f, g: M → N