## Geometry & Topology

### Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

Ulrich Koschorke

#### Abstract

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

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

Dates
Revised: 9 March 2006
Accepted: 21 April 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799737

Digital Object Identifier
doi:10.2140/gt.2006.10.619

Mathematical Reviews number (MathSciNet)
MR2240900

Zentralblatt MATH identifier
1128.55002

#### Citation

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. https://projecteuclid.org/euclid.gt/1513799737

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