A Nielsen theory for intersection numbers

Christopher McCord

Fundamenta Mathematicae (1997)

  • Volume: 152, Issue: 2, page 117-150
  • ISSN: 0016-2736

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. In this paper, the techniques of Nielsen theory are applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), is introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, NI(f,g) gives a lower bound for the number of points of intersection for all maps homotopic to f and g.

How to cite

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McCord, Christopher. "A Nielsen theory for intersection numbers." Fundamenta Mathematicae 152.2 (1997): 117-150. <http://eudml.org/doc/212202>.

@article{McCord1997,
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. In this paper, the techniques of Nielsen theory are applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), is introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, NI(f,g) gives a lower bound for the number of points of intersection for all maps homotopic to f and g.},
author = {McCord, Christopher},
journal = {Fundamenta Mathematicae},
keywords = {Nielsen number; Nielsen coincidence theory; Nielsen intersection number; Nielsen theory},
language = {eng},
number = {2},
pages = {117-150},
title = {A Nielsen theory for intersection numbers},
url = {http://eudml.org/doc/212202},
volume = {152},
year = {1997},
}

TY - JOUR
AU - McCord, Christopher
TI - A Nielsen theory for intersection numbers
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 2
SP - 117
EP - 150
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. In this paper, the techniques of Nielsen theory are applied to the study of intersections of maps. A Nielsen-type number, the Nielsen intersection number NI(f,g), is introduced, and shown to have many of the properties analogous to those of the Nielsen fixed point number. In particular, NI(f,g) gives a lower bound for the number of points of intersection for all maps homotopic to f and g.
LA - eng
KW - Nielsen number; Nielsen coincidence theory; Nielsen intersection number; Nielsen theory
UR - http://eudml.org/doc/212202
ER -

References

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  1. [1] R. Brooks, 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 removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy, Pacific J. Math. 39 (1971), 45-52. Zbl0235.55006
  4. [4] R. Dobreńko and Z. Kucharski, On the generalization of the Nielsen number, Fund. Math. 134 (1990), 1-14. Zbl0719.55002
  5. [5] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1970. 
  6. [6] P. Heath, E. Keppelmann and P. Wong, Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps, Topology Appl. 67 (1995), 133-157. Zbl0845.55004
  7. [7] M. Hirsch, Differential Topology, Springer, Berlin, 1976. Zbl0356.57001
  8. [8] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, R.I., 1983. 
  9. [9] C. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math. 154 (1992), 345-368. Zbl0766.55002
  10. [10] C. McCord, The three faces of Nielsen: comparing coincidence numbers, intersection numbers and root numbers, in preparation. 
  11. [11] J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, 1965. Zbl0161.20302
  12. [12] C.-Y. You, Fixed point classes of a fiber map, Pacific J. Math. 100 (1982), 217-241. Zbl0512.55004

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