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

100%

Wong P.

Fundamenta Mathematicae

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1991-1992

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tom 140

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nr 2

191-196

EN

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.

2

Fixed point theory for homogeneous spaces, II

100%

Wong P.

Fundamenta Mathematicae

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2005

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tom 186

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nr 2

161-175

EN

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 the hypothesis that p⁎: Hₙ(G) → Hₙ(M) is nontrivial where n = dim M. A simple formula using equivariant degree is given for the Reidemeister trace of a selfmap f: M → M.

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Equivariant Nielsen theory

100%

Wong P.

Banach Center Publications

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1999

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tom 49

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nr 1

253-258

4

Corrections to "On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem" (Fund. Math. 140 (1992), 191–196)

100%

Wong P.

Fundamenta Mathematicae

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1992

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tom 141

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nr 1

97-99

Ograniczanie wyników

3 Fundamenta Mathematicae

1 Banach Center Publications

4 Wong P.

1 2005

1 1999

2 1992

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

Fixed point theory for homogeneous spaces, II

Equivariant Nielsen theory

Corrections to "On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem" (Fund. Math. 140 (1992), 191–196)