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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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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
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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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.
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.
Coincidence free pairs of maps
"... Abstract This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loo ..."
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Abstract This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces.
SOME HOMOTOPY THEORETICAL QUESTIONS ARISING IN NIELSEN COINCIDENCE THEORY
"... Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility properties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems ..."
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Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility properties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems to be important for a more comprehensive understanding.