Let f be a map of a tree-like continuum M that sends each arc-component of M into itself. We prove that f has a fixed point. Hence every tree-like continuum has the fixed-point property for deformations (maps that are homotopic to the identity). This result answers a question of Bellamy. Our proof resembles an old argument of Brouwer involving uncountably many tangent curves. The curves used by Brouwer were originally defined by Peano. In place of these curves, we use rays that were originally defined by Borsuk.
@article{bwmeta1.element.bwnjournal-article-fmv155i2p161bwm,
author = {Charles Hagopian},
title = {The fixed-point property for deformations of tree-like continua},
journal = {Fundamenta Mathematicae},
volume = {158},
year = {1998},
pages = {161-176},
zbl = {0918.54034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv155i2p161bwm}
}
Hagopian, Charles. The fixed-point property for deformations of tree-like continua. Fundamenta Mathematicae, Tome 158 (1998) pp. 161-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv155i2p161bwm/
[00000] [A] M. A. Armstrong, Basic Topology, McGraw-Hill, London, 1979.
[00001] [B] D. P. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math. 6 (1979), 1-13. | Zbl 0447.54039
[00002] [Be] R. Bennett, Locally connected 2-cell and 2-sphere-like continua, Proc. Amer. Math. Soc. 17 (1966), 674-681. | Zbl 0139.40702
[00003] [Bi1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. | Zbl 0043.16804
[00004] [Bi2] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132. | Zbl 0174.25902
[00005] [Bo1] K. Borsuk, Sur un continu acyclique qui se laisse transformer topologiquement en lui même sans points invariants, Fund. Math. 24 (1935), 51-58. | Zbl 0010.13402
[00006] [Bo2] K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. 2 (1954), 17-20. | Zbl 0057.39103
[00007] [Br] L. E. J. Brouwer, On continuous vector distributions on surfaces, Proc. Konink. Akad. Wetensch. (Amsterdam) 11 (1909), 850-858.
[00008] [C] R. W. Conn, The engineering of magnetic fusion reactors, Scientific American 249 (4) (October, 1983), 60-71.
[00009] [Co] H. Cook, Tree-likeness of dendroids and λ-dendroids, Fund. Math. 68 (1970), 19-22. | Zbl 0203.25102
[00010] [H1] C. L. Hagopian, Fixed-point problems for disk-like continua, Amer. Math. Monthly 83 (1976), 471-473. | Zbl 0337.54027
[00011] [H2] C. L. Hagopian, Uniquely arcwise connected plane continua have the fixed-point property, Trans. Amer. Math. Soc. 248 (1979), 85-104. | Zbl 0407.54028
[00012] [H3] C. L. Hagopian, The fixed-point property for deformations of uniquely arcwise connected continua, Topology Appl. 24 (1986), 207-212. | Zbl 0606.54029
[00013] [H4] C. L. Hagopian, Fixed points of arc-component-preserving maps, Trans. Amer. Math. Soc. 306 (1988), 411-420. | Zbl 0642.54027
[00014] [H5] C. L. Hagopian, Fixed points of tree-like continua, in: Contemp. Math. 72, Amer. Math. Soc., 1988, 131-137.
[00015] [H6] C. L. Hagopian, A fixed-point theorem for tree-like continua, Topology Proc. 16 (1991), 57-62. | Zbl 0786.54043
[00016] [H7] C. L. Hagopian, Fixed-point problems in continuum theory, in: Contemp. Math. 117, Amer. Math. Soc., 1991, 79-86. | Zbl 0738.54011
[00017] [H8] C. L. Hagopian, The fixed-point property for simply-connected plane continua, Trans. Amer. Math. Soc. 348 (1996), 4525-4548. | Zbl 0878.54026
[00018] [L] S. Lefschetz, Continuous transformations of manifolds, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 90-93. | Zbl 49.0409.01
[00019] [Le] I. W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361-394.
[00020] [M] H. F. Mathis, A short proof that an isotropic antenna is impossible, Proc. Institute Radio Engineers 39 (1951), 970.
[00021] [Mi1] P. Minc, A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories, Topology Appl. 46 (1992), 99-106. | Zbl 0770.54043
[00022] [Mi2] P. Minc, A periodic point free homeomorphism of a tree-like continuum, Trans. Amer. Math. Soc. 348 (1996), 1487-1519. | Zbl 0863.54027
[00023] [OR] L. G. Oversteegen and J. T. Rogers, Jr., Fixed-point-free maps on tree-like continua, Topology Appl. 13 (1982), 85-95. | Zbl 0478.54028
[00024] [W] G. T. Whyburn, Analytic Topology, rev. ed., Amer. Math. Soc. Colloq. Publ. 28, Amer. Math. Soc., Providence, R.I., 1963. | Zbl 0117.15804
[00025] [Y] G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880-884. | Zbl 0102.37806