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

DOI: http://dx.doi.org/10.12775/TMNA.2009.007

#### 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.

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.

#### Keywords

Coincidence; fixed point; map over B; normal bordism; $\omega$-invariant; Nielsen number; Reidemeister class; Dold's index; fibration

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