Some spaces that do not have the common fixed point property
HTML articles powered by AMS MathViewer
- by John Philip Huneke and Henry H. Glover PDF
- Proc. Amer. Math. Soc. 29 (1971), 190-196 Request permission
Abstract:
For what topological spaces X do every pair of self maps of X which commute under composition have a common fixed point? No nontrivial examples of such spaces are known. Since every self map commutes with itself, X does not have this property if X does not have the fixed point property. It is shown that every completely regular Hausdorff space containing an arc does not have this property. In general, the self maps for these spaces are not surjective. The image is the arc. For surjective self maps it is shown that every topological manifold with nonnegative Euler characteristic does not have this property. An earlier counterexample for the closed interval is used in all proofs. This counterexample is due to Huneke.References
- Morton Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331–341. MR 133812, DOI 10.2307/1970177
- Robert F. Brown, Path fields on manifolds, Trans. Amer. Math. Soc. 118 (1965), 180–191. MR 173269, DOI 10.1090/S0002-9947-1965-0173269-2
- Edward Fadell, Recent results in the fixed point theory of continuous maps, Bull. Amer. Math. Soc. 76 (1970), 10–29. MR 271935, DOI 10.1090/S0002-9904-1970-12358-8
- John Philip Huneke, On common fixed points of commuting continuous functions on an interval, Trans. Amer. Math. Soc. 139 (1969), 371–381. MR 237724, DOI 10.1090/S0002-9947-1969-0237724-2 —, On commuting functions without common fixed points, Notices Amer. Math. Soc. 16 (1969), 789-790. Abstract #667-116.
- J. T. Joichi, On functions that commute with full functions and common fixed points, Nieuw Arch. Wisk. (3) 14 (1966), 247–251. MR 205245
- J. M. Kister, Microbundles are fibre bundles, Bull. Amer. Math. Soc. 69 (1963), 854–857. MR 156359, DOI 10.1090/S0002-9904-1963-11064-2
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 190-196
- MSC: Primary 54.85; Secondary 57.00
- DOI: https://doi.org/10.1090/S0002-9939-1971-0278284-4
- MathSciNet review: 0278284