Topological Methods in Nonlinear Analysis

Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index

Daciberg L. Gonçalves and Ulrich Koschorke

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Let $M \to B$, $N \to B$ be fibrations and $f_1,f_2\colon M \to N$ be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair $f_1,f_2$ over $B$ to a coincidence free pair of maps. In the special case where the two fibrations are the same and one of the maps is the identity, a weak version of our $\omega$-invariant turns out to equal Dold's fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over $B$ are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between $S^1$-bundles over $S^1$ as well as their Nielsen and Reidemeister numbers.

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Topol. Methods Nonlinear Anal., Volume 33, Number 1 (2009), 85-103.

First available in Project Euclid: 27 April 2016

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Gonçalves, Daciberg L.; Koschorke, Ulrich. Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index. Topol. Methods Nonlinear Anal. 33 (2009), no. 1, 85--103.

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