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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
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.
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
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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Cited by 2 (2 self)
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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.
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.