# A Nielsen theory for intersection numbers

Fundamenta Mathematicae (1997)

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

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topMcCord, 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

top- [1] R. Brooks, A lower bound for the Δ-Nielsen number, Trans. Amer. Math. Soc. 143 (1969), 555-564. Zbl0196.26603
- [2] R. Brooks, The number of roots of f(x) = a, Bull. Amer. Math. Soc. 76 (1970), 1050-1052. Zbl0204.23202
- [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] R. Dobreńko and Z. Kucharski, On the generalization of the Nielsen number, Fund. Math. 134 (1990), 1-14. Zbl0719.55002
- [5] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1970.
- [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] M. Hirsch, Differential Topology, Springer, Berlin, 1976. Zbl0356.57001
- [8] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, R.I., 1983.
- [9] C. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math. 154 (1992), 345-368. Zbl0766.55002
- [10] C. McCord, The three faces of Nielsen: comparing coincidence numbers, intersection numbers and root numbers, in preparation.
- [11] J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, 1965. Zbl0161.20302
- [12] C.-Y. You, Fixed point classes of a fiber map, Pacific J. Math. 100 (1982), 217-241. Zbl0512.55004

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