Nielsen coincidence theory of fibre-preserving maps and Dold's fixed point index
Keywords
Coincidence, fixed point, map over B, normal bordism, $\omega$-invariant, Nielsen number, Reidemeister class, Dold's index, fibrationAbstract
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.Downloads
Published
2009-03-01
How to Cite
1.
GONÇALVES, Daciberg L. and KOSCHORKE, Ulrich. Nielsen coincidence theory of fibre-preserving maps and Dold’s fixed point index. Topological Methods in Nonlinear Analysis. Online. 1 March 2009. Vol. 33, no. 1, pp. 85 - 103. [Accessed 29 March 2024].
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