Geometry & Topology

Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Ulrich Koschorke

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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.

Article information

Geom. Topol., Volume 10, Number 2 (2006), 619-666.

Received: 29 September 2005
Revised: 9 March 2006
Accepted: 21 April 2006
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25] 55Q25: Hopf invariants 55S35: Obstruction theory 57R90: Other types of cobordism [See also 55N22]
Secondary: 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55P35: Loop spaces 55Q40: Homotopy groups of spheres

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


Koschorke, Ulrich. Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms. Geom. Topol. 10 (2006), no. 2, 619--666. doi:10.2140/gt.2006.10.619.

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