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On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem

Peter Wong — 1992

Fundamenta Mathematicae

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 = M ˜ 2 / K where 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...

Fixed point theory for homogeneous spaces, II

Peter Wong — 2005

Fundamenta Mathematicae

Let G be a compact connected Lie group, K a closed subgroup and M = G/K the homogeneous space of right cosets. Suppose that M is orientable. We show that for any selfmap f: M → M, L(f) = 0 ⇒ N(f) = 0 and L(f) ≠ 0 ⇒ N(f) = R(f) where L(f), N(f), and R(f) denote the Lefschetz, Nielsen, and Reidemeister numbers of f, respectively. In particular, this implies that the Lefschetz number is a complete invariant, i.e., L(f) = 0 iff f is deformable to be fixed point free. This was previously known under...

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