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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.
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.