Lefschetz coincidence class for several maps

Article

Abstract

The aim of this paper is to define a Lefschetz coincidence class for several maps. More specifically, for maps \({f_{1}, \ldots , f_{k} : X \rightarrow N}\) from a topological space X into a connected closed n-manifold (even nonorientable) N, a cohomological class
$$L(f_{1}, \ldots , f_{k}) \in H^{n(k-1)}(X; (f_{1}, \ldots , f_{k}) ^{\ast} (R \times \Gamma^{\ast}_{N} \times \ldots \times \Gamma^{\ast} _{N}))$$
is defined in such a way that \({L(f_{1}, \ldots , f_{k}) \neq 0}\) implies that the set of coincidences
$${\rm Coin}(f_{1}, \ldots , f_{k}) = \{x \in X\,|\,f_{1}(x) = \ldots = f_{k}(x)\}$$
is nonempty.

Mathematics Subject Classification

Primary 55M20 Secondary 54H25 

Keywords

Coincidence point Lefschetz coincidence number 

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References

  1. 1.
    Biasi C., Libardi A.K.M., Monis T.F.M.: The Lefschetz coincidence class of p maps. Forum Math. 27, 1717–1728 (2015)CrossRefMathSciNetGoogle Scholar
  2. 2.
    G. E. Bredon, Sheaf Theory. 2nd ed., Springer-Verlag, New York, 1997.Google Scholar
  3. 3.
    Gonçalves D. L., Jezierski J.: Lefschetz coincidence formula on nonorientable manifolds. Fund. Math. 153, 1–23 (1997)MathSciNetMATHGoogle Scholar
  4. 4.
    M. J. Greenberg and J. R. Harper, Algebraic Topology. A first Course. Benjamin/ Cummings Publishing Co., Inc., Reading, Mass., 1981.Google Scholar
  5. 5.
    Spanier E.: Algebraic Topology. McGraw–Hill, New York (1966)MATHGoogle Scholar
  6. 6.
    E. Spanier, Duality in topological manifolds. In: Colloque de Topologie Tenu á Bruxelles (Brussels, 1964), Librairie Universitaire, Louvain, 1966, 91–111.Google Scholar
  7. 7.
    Spanier E.: Singular homology and cohomology with local coefficients and duality for manifolds. Pacific J. Math. 160, 165–200 (1993)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    J. W. Vick, Homology Theory. An Introduction to Algebraic Topology. 2nd ed., Springer-Verlag, New York, 1994.Google Scholar
  9. 9.
    Whitehead G.: Elements of Homotopy Theory. Springer-Verlag, New York (1978)CrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsIGCE, UNESP – Universidade Estadual PaulistaRio Claro/SPBrazil
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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