## Topological Methods in Nonlinear Analysis

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

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 27 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461782241

Mathematical Reviews number (MathSciNet)
MR2512956

Zentralblatt MATH identifier
1178.55002

#### Citation

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. https://projecteuclid.org/euclid.tmna/1461782241

#### References

• A. Dold, The fixed point index of fibre-preserving maps , Inventiones math., 25 (1974), 281–297 \ref
• A. Dold and D. L. Gonçalves, Self-coincidence of fibre maps , Osaka J. Math., 42 (2005), 291–307 \ref
• D. L. Gonçalves and M. Kelly, Coincidence properties for maps from the torus to the Klein bottle , Chinese Ann. Math. Ser. B, to appear \ref
• D. L. Gonçalves and D. Randall, Self-coincidence of maps from $S^q$-bundles over $S^n$ to $S^n$ , Bol. Soc. Mat. Mexicana (3), 10 , Special issue in honor of Francisco “Figo” González Acuña, 3 serie (2004), 181–192 \ref ––––, Self-coincidence of mappings between spheres and the strong Kervaire invariant one problem , C. R. Acad. Sci. Paris Sér. I, 342 (2006), 511–513 \ref
• J. Jezierski, The Nielsen relation for fibre maps , Bulletin de l'academie Polonaise des Sciences, XXX, no. 5-6 (1982), 277–282 \ref
• U. Koschorke, Vector Fields and Other Vector Bundle Morphisms –- a Singularity Approach, Lecture Notes in Math., 847 , Springer–Verlag, Berlin, Heidelberg, New York (1981) \ref ––––, Self-coincidences in higher codimensions , J. Reine Angew. Math., 576 (2004), 1–10 \ref ––––, Nielsen coincidence theory in arbitrary codimensions, , J. Reine Angew. Math., 598 (2006), 211–236 \ref ––––, Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms , Geometry and Topology, 10 (2006), 619–665 \ref ––––, Geometric and homotopy theoretic methods in Nielsen coincidence theory , Fixed Point Theory and Applications, Article ID 84093 (2006), 1–15 \ref ––––, Selfcoincidences and roots in Nielsen theory , J. Fixed Point Theory Appl., 2 (2007), 241–259 \ref
• G. Whitehead, Elements of Homotopy Theory, Springer–Verlag, Berlin, Heidelberg, New York (1978)