Volume 14, issue 3 (2014)

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Nielsen coincidence numbers, Hopf invariants and spherical space forms

Ulrich Koschorke

Algebraic & Geometric Topology 14 (2014) 1541–1575
Abstract

Given two maps between smooth manifolds, the obstruction to removing their coincidences (via homotopies) is measured by the minimum numbers. In order to determine them we introduce and study an infinite hierarchy of Nielsen numbers ${N}_{i}$, $i=0,1,\dots ,\infty$. They approximate the minimum numbers from below with decreasing accuracy, but they are (in principle) more easily computable as $i$ grows. If the domain and the target manifold have the same dimension (eg in the fixed point setting) all these Nielsen numbers agree with the classical definition. However, in general they can be quite distinct.

While our approach is very geometric, the computations use the techniques of homotopy theory and, in particular, all versions of Hopf invariants (à la Ganea, Hilton or James). As an illustration we determine all Nielsen numbers and minimum numbers for pairs of maps from spheres to spherical space forms. Maps into even dimensional real projective spaces turn out to produce particularly interesting coincidence phenomena.

 Dedicated to Karl-Otto Stöhr on the occasion of his 70th birthday.
Keywords
coincidence, Nielsen number, minimum number, Hopf invariants, spherical space forms
Mathematical Subject Classification 2010
Primary: 55M20
Secondary: 55Q25, 55Q40
Publication
Revised: 22 August 2013
Accepted: 29 August 2013
Published: 7 April 2014
Authors
 Ulrich Koschorke Department Mathematik Universität Siegen Emmy Noether Campus Walter-Flex-Str. 3 D-57068 Siegen Germany