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

Peter Wong

Fundamenta Mathematicae (1992)

  • Volume: 140, Issue: 2, page 191-196
  • ISSN: 0016-2736

Abstract

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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 = 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 and g are deformable to be coincidence free if the Lefschetz coincidence number L(f,g) vanishes.

How to cite

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Wong, Peter. "On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem." Fundamenta Mathematicae 140.2 (1992): 191-196. <http://eudml.org/doc/211938>.

@article{Wong1992,
abstract = {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.},
author = {Wong, Peter},
journal = {Fundamenta Mathematicae},
keywords = {fixed points; coincidences; roots; Lefschetz number; Nielsen number; Nielsen equivalence relation; coincidence index; Nielsen coincidence class; connected simply connected topological group; discrete subgroup; Lefschetz coincidence number},
language = {eng},
number = {2},
pages = {191-196},
title = {On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem},
url = {http://eudml.org/doc/211938},
volume = {140},
year = {1992},
}

TY - JOUR
AU - Wong, Peter
TI - On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem
JO - Fundamenta Mathematicae
PY - 1992
VL - 140
IS - 2
SP - 191
EP - 196
AB - 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.
LA - eng
KW - fixed points; coincidences; roots; Lefschetz number; Nielsen number; Nielsen equivalence relation; coincidence index; Nielsen coincidence class; connected simply connected topological group; discrete subgroup; Lefschetz coincidence number
UR - http://eudml.org/doc/211938
ER -

References

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  1. [1] R. Brooks, Certain subgroups of the fundamental group and the number of roots of f(x)=a, Amer. J. Math. 95 (1973), 720-728. Zbl0319.55015
  2. [2] R. Brooks and R. Brown, A lower bound for the Δ-Nielsen number, Trans. Amer. Math. Soc. 143 (1969), 555-564. Zbl0196.26603
  3. [3] R. Brooks and P. Wong, On changing fixed points and coincidences to roots, Proc. Amer. Math. Soc., to appear. Zbl0779.55001
  4. [4] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott & Foresman, Glenview, Ill., 1971. Zbl0216.19601
  5. [5] A. Dold, Lectures on Algebraic Topology, Springer, Berlin 1972. 
  6. [6] E. Fadell and S. Husseini, Local fixed point index theory for non simply connected manifolds, Illinois J. Math. 25 (1981), 673-699. Zbl0469.55004
  7. [7] J. Jezierski, The Nielsen number product formula for coincidences, Fund. Math. 134 (1989), 183-212. Zbl0715.55002
  8. [8] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., 1982. 
  9. [9] T. Kiang, The Theory of Fixed Point Classes, Springer, Berlin 1989. Zbl0676.55001
  10. [10] C. McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, Topology Appl., to appear. Zbl0748.55001
  11. [11] H. Schirmer, Mindestzahlen von Koinzidenzpunkten, J. Reine Angew. Math. 194 (1955), 21-39. 
  12. [12] J. Vick, Homology Theory, Academic Press, New York 1973. 

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