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1
Content available remote

The Reidemeister trace and the calculation of the Nielsen number

100%
Hart E. L.
Banach Center Publications
|
1999
|
tom 49
|
nr 1
151-157
2
Content available remote

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

100%
Graff G., Kaczkowska A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2160-2172
EN
Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.
3
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On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem

88%
Wong P.
Fundamenta Mathematicae
|
1991-1992
|
tom 140
|
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.
4
Content available remote

Epsilon Nielsen coincidence theory

88%
Fenille M.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1337-1348
EN
We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and idY is its identity map, then µ ∈(f, idY) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ∈(f, g) as a lower bound for µ ∈(f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.
5
Content available remote

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

88%
Wong P.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 1
97-99
6
Content available remote

Equivariant Nielsen theory

75%
Wong P.
Banach Center Publications
|
1999
|
tom 49
|
nr 1
253-258
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