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Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Ulrich Koschorke

Geometry & Topology 10 (2006) 619–666

arXiv: math/0606025


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.

coincidence manifold, normal bordism, path space, Nielsen number, Ganea-Hopf invariant
Mathematical Subject Classification 2000
Primary: 55M20, 55Q25, 55S35, 57R90
Secondary: 55N22, 55P35, 55Q40
Forward citations
Received: 29 September 2005
Revised: 9 March 2006
Accepted: 21 April 2006
Published: 24 May 2006
Proposed: Shigeyuki Morita
Seconded: Wolfgang Lück, Tom Goodwillie
Ulrich Koschorke
Universität Siegen
Emmy Noether Campus
Walter-Flex-Str. 3
D-57068 Siegen