Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Koschorke, Ulrich

doi:10.2140/gt.2006.10.619

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Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Ulrich Koschorke

Geometry & Topology 10 (2006) 619–666

DOI: 10.2140/gt.2006.10.619

arXiv: math/0606025

Abstract

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs $(f_{1}, f_{2})$ of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers $M C C (f_{1}, f_{2})$ (and $M C (f_{1}, f_{2})$ , resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to $(f_{1}, f_{2})$ . Furthermore, we deduce finiteness conditions for $M C (f_{1}, f_{2})$ . 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 $E (f_{1}, f_{2})$ 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 $M C$ .

Keywords

coincidence manifold, normal bordism, path space, Nielsen number, Ganea-Hopf invariant

Mathematical Subject Classification 2000

Primary: 55M20, 55Q25, 55S35, 57R90

Secondary: 55N22, 55P35, 55Q40

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Publication

Received: 29 September 2005

Revised: 9 March 2006

Accepted: 21 April 2006

Published: 24 May 2006

Proposed: Shigeyuki Morita

Seconded: Wolfgang Lück, Tom Goodwillie

Authors

Ulrich Koschorke
Universität Siegen Emmy Noether Campus Walter-Flex-Str. 3 D-57068 Siegen Germany
http://www.math.uni-siegen.de/topology/

Volume 10, issue 2 (2006)