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.
@article{bwmeta1.element.bwnjournal-article-fmv152i2p117bwm,
author = {Christopher McCord},
title = {A Nielsen theory for intersection numbers},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {117-150},
zbl = {0882.55001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv152i2p117bwm}
}
McCord, Christopher. A Nielsen theory for intersection numbers. Fundamenta Mathematicae, Tome 154 (1997) pp. 117-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv152i2p117bwm/
[00000] [1] R. Brooks, A lower bound for the Δ-Nielsen number, Trans. Amer. Math. Soc. 143 (1969), 555-564. | Zbl 0196.26603
[00001] [2] R. Brooks, The number of roots of f(x) = a, Bull. Amer. Math. Soc. 76 (1970), 1050-1052. | Zbl 0204.23202
[00002] [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. | Zbl 0235.55006
[00003] [4] R. Dobreńko and Z. Kucharski, On the generalization of the Nielsen number, Fund. Math. 134 (1990), 1-14. | Zbl 0719.55002
[00004] [5] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1970.
[00005] [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. | Zbl 0845.55004
[00006] [7] M. Hirsch, Differential Topology, Springer, Berlin, 1976. | Zbl 0356.57001
[00007] [8] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, R.I., 1983.
[00008] [9] C. McCord, Estimating Nielsen numbers on infrasolvmanifolds, Pacific J. Math. 154 (1992), 345-368. | Zbl 0766.55002
[00009] [10] C. McCord, The three faces of Nielsen: comparing coincidence numbers, intersection numbers and root numbers, in preparation.
[00010] [11] J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, 1965. | Zbl 0161.20302
[00011] [12] C.-Y. You, Fixed point classes of a fiber map, Pacific J. Math. 100 (1982), 217-241. | Zbl 0512.55004