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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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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.
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
GRADED POISSON ALGEBRAS ON BORDISM GROUPS OF GARLANDS AND THEIR APPLICATIONS
, 2006
"... Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of Ngarlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our prev ..."
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Cited by 4 (1 self)
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Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of Ngarlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our previous work with Rudyak we introduced a rich algebra structure on the bordism group Ω∗(GN,M). In this work we introduce the operations ⋆ and [·, ·] on the tensor product of Q and a certain bordism group ̂Ω∗(GN,M). For N consisting of odddimensional manifolds, these operations make ̂Ω∗(GN,M) ⊗ Q into a graded Poisson algebra (Gerstenhaberlike algebra). For N consisting of evendimensional manifolds, ⋆ satisfies a graded Leibniz rule with respect to [·, ·], but [·, ·] does not satisfy a graded Jacobi identity. The mod 2analogue of [·, ·] for oneelement sets N was previously constructed in our preprint with Rudyak. For N = {S 1} and a surface F 2, the subalgebra ̂ Ω0(G {S 1},F 2) ⊗ Q of our algebra is related to the GoldmanTuraev algebra of loops on a surface and to the AndersenMattesReshetikhin Poisson algebra of chorddiagrams. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes ̂ δ1, ̂ δ2 of free loops on F 2, provided that ̂ δ1, ̂ δ2 do not contain powers of the same element of π1(F 2).
Kervaire invariants and selfcoincidences
, 2007
"... Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In th ..."
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Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m = 2n − 2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) are obtained in the next seven dimension ranges (when 1 < m − 2n + 3 8). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126stem or not.
Nielsen coincidence theory of fibrepreserving maps and Dold’s fixed point index Topological Methods in Nonlinear Analysis
 Journal of the Juliusz Schauder Center
"... Dedicated to Albrecht Dold on the occasion of his 80th birthday Abstract. Let M → B, N → B be fibrations and f1, f2: M → N be a pair of fibrepreserving 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 ..."
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Dedicated to Albrecht Dold on the occasion of his 80th birthday Abstract. Let M → B, N → B be fibrations and f1, f2: M → N be a pair of fibrepreserving 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 ωinvariant turns out to equal Dold’s fixed point index of fibrepreserving 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 S1−bundles over S1 as well as their Nielsen and Reidemeister numbers.
LINKING AND COINCIDENCE INVARIANTS
, 2004
"... Abstract. Given a link map f into a manifold of the form Q = N ×R, when can it be deformed to an “unlinked ” position (in some sense, e.g. where its components map to disjoint Rlevels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obs ..."
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Abstract. Given a link map f into a manifold of the form Q = N ×R, when can it be deformed to an “unlinked ” position (in some sense, e.g. where its components map to disjoint Rlevels)? Using the language of normal bordism theory as well as the path space approach of Hatcher and Quinn we define obstructions ˜ωε(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 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.
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.