Index zero fixed points and 2-complexes with local separating points
Keywords
Minimal map, fixed point, local separating point, indexAbstract
We study the situation of an isolated fixed point with local index zero at a local separating point in a $2$-complex. This fixed point sometimes can be removed and sometimes not, either locally or globally. Criteria are given for local removability in dimension at most two. Results are applied to finding fixed point minimal models on $S^2\vee S^2$ and $S^1 \vee S^2$. A non-existence result is given in the case of a wedge in which both factors are surfaces with non-positive Euler characteristic.References
R.F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman, 1971.
A. Dold, Fixed point index and fixed point theorem for Euclidean neighbourhood retracts, Topology 4 (1965), 1–8.
D.L. Gonçalves and M.R. Kelly, Fixed point index bounds for self-maps on closed surfaces, Bull Belg. Math. Soc. Simon Stevin 24 (2017), 673–688.
D.L. Gonçalves and M.R. Kelly, Index bounds for maps defined on a wedge of surfaces, JP Journal of Geometry and Topology 21 (2018), no. 3, 255–264.
B. Jiang, On the least number of fixed points, Amer. J. Math. 102 (1980), 749–763.
B. Jiang, Lectures on Nielsen Fixed Point Theory, American Mathematical Society, 1983.
B. Jiang, Fixed points and braids, Inv. Math. 75 (1984), 69–74.
B. Jiang, Fixed points and braids II, Math. Ann. 272 (1985), 249–256.
B. Jiang, A primer of Nielsen fixed point theory, Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, pp. 617–647.
N. Khamsemanan and S.W. Kim, Estimating Nielsen numbers on wedge product spaces, Fixed Point Theory Appl. 2007, Art. ID 83420, 16 pp.
T.-h. Kiang, The Theory of Fixed Point Classes, Springer–Verlag, Berlin, 1989.
J. Llibre and A. Nunes, Minimum number of fixed points for maps of the figure eight space, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), 1795–1802.
O.M. Neto and N.C.L. Penteado, Representing homotopy classes by maps with minimality root properties, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), No. 5, 1005–1015.
G.H. Shi, On the least number of fixed points and Nielsen numbers, Acta Math. Sinica 16 (1966), 223–232.
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