Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of fixed points in cases where continuous map extensions behave differently from smooth ones. In the appendix it is shown that a subset of a smooth manifold can be realized as the fixed point set of a smooth map in a given homotopy class if and only if it can be realized as the fixed point set of a continuous one. A special case of this result is used in a proof of the paper.
@article{bwmeta1.element.bwnjournal-article-fmv141i1p31bwm,
author = {Helga Schirmer},
title = {Nielsen theory of transversal fixed point sets (with an appendix: $C^$\infty$$ and C0 fixed point sets are the same, by R. E. Greene)},
journal = {Fundamenta Mathematicae},
volume = {141},
year = {1992},
pages = {31-59},
zbl = {0812.55002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv141i1p31bwm}
}
Schirmer, Helga. Nielsen theory of transversal fixed point sets (with an appendix: $C^∞$ and C0 fixed point sets are the same, by R. E. Greene). Fundamenta Mathematicae, Tome 141 (1992) pp. 31-59. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv141i1p31bwm/
[00000] [1] D. V. Anasov, The Nielsen numbers of nil-manifolds, Uspekhi Mat. Nauk 40 (1985), 133-134; Russian Math. Surveys 40 (1985), 149-150.
[00001] [2] R. F. Brown, R. E. Greene and H. Schirmer, Fixed points of map extensions, in: Topological Fixed Point Theory and Applications (Proc. Tianjin 1988), Lecture Notes in Math. 1411, Springer, Berlin 1989, 24-45.
[00002] [3] R. E. Greene and H. Wu, approximations of convex, subharmonic and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), 47-84. | Zbl 0415.31001
[00003] [4] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.
[00004] [5] B. Jiang, Fixed point classes from a differentiable viewpoint, in: Fixed Point Theory (Proc. Sherbrooke, Québec, 1980), Lecture Notes in Math. 886, Springer, Berlin 1981, 163-170.
[00005] [6] B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, R.I., 1983. | Zbl 0512.55003
[00006] [7] B. Jiang, Fixed points and braids. II, Math. Ann. 272 (1985), 249-256. | Zbl 0617.55001
[00007] [8] J. Munkres, Elementary Differential Topology, Princeton Univ. Press, 1966.
[00008] [9] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986), 459-473. | Zbl 0553.55001
[00009] [10] H. Schirmer, On the location of fixed points on pairs of spaces, Topology Appl. 30 (1988), 253-266. | Zbl 0664.55003
[00010] [11] H. Schirmer, Fixed point sets in a prescribed homotopy class, ibid., to appear.
[00011] [12] X. Zhao, A relative Nielsen number for the complement, in: Topological Fixed Point Theory and Applications (Proc. Tianjin 1988), Lecture Notes in Math. 1411, Springer, Berlin 1989, 189-199.
[00012] [1] R. F. Brown, R. E. Greene and H. Schirmer, Fixed points of map extensions, in: Topological Fixed Point Theory and Applications (Proc. Tianjin 1988), Lecture Notes in Math. 1411, Springer, Berlin 1989, 24-45.
[00013] [2] R. E. Greene, Complete metrics of bounded curvature on noncompact manifolds, Arch. Math. (Basel) 31 (1978), 89-95. | Zbl 0373.53018