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Transactions of the American Mathematical Society
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
Published electronically: September 2, 2004
MathSciNet review: 2146635
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03533-0
PII: S 0002-9947(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