Geodesic
On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem
Fundamenta Mathematicae, Tome 140 (1991) no. 2, pp. 191-196

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Let $f,g:M_1 → M_2$ be maps where $M_1$ and $M_2$ are connected triangulable oriented n-manifolds so that the set of coincidences $C_{f,g}= {x ∈ M_1 | f(x)=g(x)}$ is compact in $M_1$. We define a Nielsen equivalence relation on $C_{f,g}$ and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if $M_2= \widetilde M_2/K$ where $\widetilde M_2$ is a connected simply connected topological group and K is a discrete subgroup then all the Nielsen coincidence classes of f and g have the same coincidence index. In particular, when $M_1$ and $M_2$ are compact, f and g are deformable to be coincidence free if the Lefschetz coincidence number L(f,g) vanishes.
DOI : 10.4064/fm-140-2-191-196
Keywords: fixed points, coincidences, roots, Lefschetz number, Nielsen number 
Peter Wong. On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem. Fundamenta Mathematicae, Tome 140 (1991) no. 2, pp. 191-196. doi: 10.4064/fm-140-2-191-196
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