Wecken theorems for Nielsen intersection theory

• Volume: 49, Issue: 1, page 235-252
• ISSN: 0137-6934

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Abstract

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Nielsen theory, originally developed as a homotopy-theoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. Recently, the techniques of Nielsen theory have been applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), was introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, it is a homotopy-invariant lower bound for the number of intersections of a pair of maps. The question of whether or not this lower bound is sharp can be thought of as the Wecken problem for intersection theory. In this paper, the Wecken problem for intersections is considered, and some Wecken theorems are proved.

How to cite

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McCord, Christopher. "Wecken theorems for Nielsen intersection theory." Banach Center Publications 49.1 (1999): 235-252. <http://eudml.org/doc/208964>.

@article{McCord1999,
abstract = {Nielsen theory, originally developed as a homotopy-theoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. Recently, the techniques of Nielsen theory have been applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), was introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, it is a homotopy-invariant lower bound for the number of intersections of a pair of maps. The question of whether or not this lower bound is sharp can be thought of as the Wecken problem for intersection theory. In this paper, the Wecken problem for intersections is considered, and some Wecken theorems are proved.},
author = {McCord, Christopher},
journal = {Banach Center Publications},
keywords = {intersection set; Nielsen number; Wecken property},
language = {eng},
number = {1},
pages = {235-252},
title = {Wecken theorems for Nielsen intersection theory},
url = {http://eudml.org/doc/208964},
volume = {49},
year = {1999},
}

TY - JOUR
AU - McCord, Christopher
TI - Wecken theorems for Nielsen intersection theory
JO - Banach Center Publications
PY - 1999
VL - 49
IS - 1
SP - 235
EP - 252
AB - Nielsen theory, originally developed as a homotopy-theoretic approach to fixed point theory, has been translated and extended to various other problems, such as the study of periodic points, coincidence points and roots. Recently, the techniques of Nielsen theory have been applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), was introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, it is a homotopy-invariant lower bound for the number of intersections of a pair of maps. The question of whether or not this lower bound is sharp can be thought of as the Wecken problem for intersection theory. In this paper, the Wecken problem for intersections is considered, and some Wecken theorems are proved.
LA - eng
KW - intersection set; Nielsen number; Wecken property
UR - http://eudml.org/doc/208964
ER -

References

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1. [1] R. Brooks and R. F. Brown, A lower bound for the Δ-Nielsen number, Trans. Amer. Math. Soc. 143 (1969), 555-564. Zbl0196.26603
2. [2] R. Brooks, The number of roots of f(x) = a, Bull. Amer. Math. Soc. 76 (1970), 1050-1052. Zbl0204.23202
3. [3] R. Brooks, On the sharpness of the ${\Delta }_{2}$ and ${\Delta }_{1}$ Nielsen numbers, J. Reine Angew. Math. 259 (1973), 101-108.
4. [4] R. Dobreńko and Z. Kucharski, On the generalization of the Nielsen number, Fund. Math. 134 (1990), 1-14. Zbl0719.55002
5. [5] R. Dobreńko and J. Jezierski, The coincidence Nielsen theory on non-orientable manifolds, Rocky Mountain J. Math. 23 (1993), 67-85. Zbl0787.55003
6. [6] A. Fathi, F. Laudenbach et V. Poénaru, Travaux de Thurston sur les surfaces, Séminaire Orsay, Astérisque 66-67 (1979).
7. [7] M. Hirsch, Differential Topology, Springer-Verlag, Berlin, 1976. Zbl0356.57001
8. [8] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, RI, 1983.
9. [9] B. Jiang, Fixed points and braids, Invent. Math. 75 (1984), 69-74. Zbl0565.55005
10. [10] C. McCord, A Nielsen theory for intersection numbers, Fund. Math. 152 (1997), 117-150. Zbl0882.55001
11. [11] C. McCord, The three faces of Nielsen: comparing coincidence numbers, intersection numbers and root numbers, in preparation.
12. [12] J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, 1965. Zbl0161.20302

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