Abstract
In this paper we develop a fixed point index theory for symmetric product mappings of ENR-spaces. For such mappings we show that an index can be defined which is an extension of the usual integer-valued fixed point index. Further, we show that the classical properties of the index hold in this setting: The index is additive, multiplicative and commutative, the index is preserving under homotopy, and finally, the index is equal to the Lefschetz number as defined by Maxwell [9].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bredon, G. E.: Introduction to Compact Transformation Groups, New York-London: Academic Press, 1972.
Brown, R. F.: The Lefschetz Fixed Point Theorem, Glenview, Illinois: Scott, Forseman and Company, 1971.
Dold, A.: Fixed Point Index and Fixed Point Theorem for Euclidean Neighborhood Retracts, Topology 4 (1965), 1–8.
—: Lectures on Algebraic Topology, Berlin-Heidelberg-New York: Springer-Verlag, second edition, 1980.
Hu, Sze-Tsen: Theory of Retracts, Detroit: Wayne State University Press, 1965.
Kwasik, S.: Indeks Koincydencji Odwzorowan W Przestrzenie Orbit Rozmaitosci, Preprint (1979).
Masih, S.: Fixed points of symmetric product mappings of polyhedra and metric absolute neighborhood retracts, Fund. Math. 80 (1973), 149–156.
—: On the fixed point index and Nielsen fixed point theorem for symmetric product mappings, Fund. Math. 102 (1979), 143–158.
Maxwell, C. N.: Fixed Points of Symmetric Product Mappings, Proc. Amer. Math. Soc. 8 (1957), 808–815.
—: The Degree of Multiple-Valued Maps of Spheres, Springer Lecture Notes 664 (1977), 123–141.
Milnor, J. W.: Topology from the Differentiable View Point, Charlottesville: University Press of Virginia, 1965.
Park, S.: Degrees of Symmetric Product Maps of Spheres and their Applications, Proc. Coll. Natur. Sci., SNU, Vol. 1, No. 1 (1976), 1–20.
Vick, J. W.: Homology Theory, New York-London, Academic Press, 1973.
Vora, C.: Fixed points of certain symmetric product mappings of metric manifolds, Fund. Math. 85 (1974), 19–24.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rallis, N. A fixed point index theory for symmetric product mappings. Manuscripta Math 44, 279–308 (1983). https://doi.org/10.1007/BF01166084
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01166084