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Minimizing coincidence numbers of maps into projective spaces

Ulrich Koschorke

Geometry & Topology Monographs 14 (2008) 373–391

DOI: 10.2140/gtm.2008.14.373

arXiv: math.AT/0606024


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 Strong Kervaire Invariant One Problem.


coincidence, Nielsen number, minimum coincidence number, pathspace, projective space, Kervaire invariant

Mathematical Subject Classification

Primary: 55M20

Secondary: 55Q40, 57R22


Received: 21 March 2006
Accepted: 26 January 2007
Published: 29 April 2008

Ulrich Koschorke
Universität Siegen
Emmy Noether Campus
Walter-Flex-Str. 3
D-57068 Siegen