Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fixed point index in symmetric products


Author: José M. Salazar
Journal: Trans. Amer. Math. Soc. 357 (2005), 3493-3508
MSC (2000): Primary 54H20, 54H25
DOI: https://doi.org/10.1090/S0002-9947-04-03533-0
Published electronically: September 2, 2004
MathSciNet review: 2146635
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $U$ be an open subset of a locally compact metric ANR $X$ and let $f:U \rightarrow X$ be a continuous map. In this paper we study the fixed point index of the map that $f$ induces in the $n$-symmetric product of $X$, $F_{n}(X)$. This index can detect the existence of periodic orbits of period $\leq n$ of $f$, and it can be used to obtain the Euler characteristic of the $n$-symmetric product of a manifold $X$, $\chi(F_{n}(X))$. We compute $\chi(F_{n}(X))$ for all orientable compact surfaces without boundary.


References [Enhancements On Off] (What's this?)

  • 1. M. Aguilar, S. Gitler, C. Prieto, Topología algebraica: un enfoque homotópico. McGraw Hill, México 1998; English transl., Springer-Verlag, New York, 2002. MR 2003c:53001
  • 2. K. Borsuk, On the third symmetric potency of the circunference, Fund. Math. 36 (1949) 235-244. MR 12:42a
  • 3. K. Borsuk, S. Ulam, On symmetric products of topological spaces, Bull. Amer. Math. Soc. 37 (1931) 875-882.
  • 4. R. Bott, On the third symmetric potency of $S_{1}$, Fund. Math. 39 (1952) 364-368. MR 14:1003e
  • 5. R. F. Brown, The Lefschetz fixed point theorem. Scott, Foresman and Company (1971) Glenview, Illinois. MR 14:1003e
  • 6. D. W. Curtis, Hyperspaces of noncompact metric spaces. Compositio Math. 40, 2 (1980) 139-152. MR 81c:54009
  • 7. E. N. Dancer, Degree theory on convex sets and applications to bifurcation, in Calculus of variations and partial differential equations. Edited by G. Buttazzo, A. Marino, M. K. V. Murthy. Springer-Verlag, Berlin, Heidelberg 2000 (pages 185-225). MR 2002d:49002
  • 8. R. L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley, 1987. MR 91a:58114
  • 9. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology, 4 (1965), 1-8. MR 33:1850
  • 10. R. W. Easton, Geometric methods for discrete dynamical systems, Oxford University Press, 1998. MR 2000e:37002
  • 11. A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANR's, Bull. Soc. Math. France 100 (1972) 209-228. MR 46:8213
  • 12. A. Illanes, Multicoherence of symmetric products, An. Inst. Mat. Univ. Nac. Autónoma México 25 (1985) 11-24. MR 87k:54053a
  • 13. A. Illanes, S. Macías, S. Nadler, Symmetric products and Q-manifolds, Contemporary Math. 246 (1999) 137-141. MR 2001b:54011
  • 14. P. Le Calvez, J. C. Yoccoz, Un théorème d'indice pour les homéomorphismes du plan au voisinage d'un point fixe. Annals of Math. 146 (1997) 241-293. MR 99a:58129
  • 15. S. Macías, On symmetric products of continua, Topology and its Applic. 92 (1999) 173-182. MR 2000a:54009
  • 16. S. Masih, Fixed points of symmetric product mappings of polyhedra and metric absolute neighborhood retracts, Fund. Math. 80 (1973) 149-156. MR 51:6783
  • 17. S. Masih, On the fixed point index and the Nielsen fixed point theorem of symmetric product mappings, Fund. Math. 102 (1979)143-158. MR 80f:55002
  • 18. R. Molski, On symmetric products, Fund. Math. 44 (1957) 165-170. MR 19:1186e
  • 19. N. To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), 243-254. MR 86d:54018
  • 20. R. D. Nussbaum, Generalizing the fixed point index, Math. Ann. 228 (1977), 259-278. MR 55:13461
  • 21. R. D. Nussbaum, The fixed point index and some applications, Séminaire de Mathématiques supérieures, vol. 94, Les presses de L'Université de Montréal, 1985. MR 87a:47085
  • 22. N. Rallis, A fixed point index theory for symmetric product mappings, Manuscripta Math. 44 (1983) 279-308. MR 85g:55005
  • 23. F. R. Ruiz del Portal, J. M. Salazar, Fixed point index in hyperspaces: A Conley-type index for discrete semidynamical systems, J. London Math. Soc. (2) 64 (2001) 191-204. MR 2002e:54023
  • 24. F. R. Ruiz del Portal, J. M. Salazar, Fixed point index of iterations of local homeomorphisms of the plane: A Conley index approach, Topology 41 (2002) 1199-1212. MR 2003f:37022
  • 25. J. M. Salazar, Indice de punto fijo en hiperespacios e índice de Conley, Thesis (2001) Universidad Complutense de Madrid.
  • 26. R. M. Schori, Hyperspaces and symmetric products of topological spaces, Fund. Math. 63 (1968) 77-88. MR 38:661
  • 27. W. Wu, Note sur les produits essentiels symétriques des espaces topologiques, C. R. Acad. Sci. Paris 224 (1947) 1139-1141. MR 8:479g

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54H20, 54H25

Retrieve articles in all journals with MSC (2000): 54H20, 54H25


Additional Information

José M. Salazar
Affiliation: Departamento de Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid 28871, Spain
Email: josem.salazar@uah.es

DOI: https://doi.org/10.1090/S0002-9947-04-03533-0
Keywords: Fixed point index, hyperspaces, symmetric product, semidynamical systems
Received by editor(s): May 23, 2003
Received by editor(s) in revised form: October 22, 2003
Published electronically: September 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society