Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 239–244 | Cite as

Coincidence Wecken Property for Nilmanifolds

  • Daciberg GonçalvesEmail author
  • Peter Wong


Let \(f,g : X \rightarrow Y\) be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with \(X \geq \rm{dim}\it{Y}\). In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ~ f′ so that \(C(f',g)= \{x \in X|f'(x)=g(x)\}\) is empty.


Nielsen coincidence theory nilmanifolds 

MR(2010) Subject Classification

55M20 22E25 


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  1. [1]
    Brooks, R. B. S.: On removing coincidences of two maps when only one, rather than both, of them may be deformed by a homotopy. Pacific J. Math., 40, 45–52 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Brooks, R. B. S.: On the sharpness of the Δ2 and Δ1 Nielsen numbers. J. Reine Angew. Math., 259, 101–108 (1973)MathSciNetzbMATHGoogle Scholar
  3. [3]
    Gonçalves, D., Jezierski, J., Wong, P.: Obstruction theory and coincidences in positive codimension. Acta Math. Sin. Engl. Ser., 22(5), 1591–1602 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Gonçalves, D., Wong, P.: Wecken property for roots. Proc. Amer. Math. Soc., 133(9), 2779–2782 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Gonçalves, D., Wong, P.: Obstruction theory and coincidences of maps between nilmanifolds. Archiv Math., 84, 568–576 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Gonçalves, D., Wong, P.: Nilmanifolds are Jiang-type spaces for coincidences. Forum Math., 13, 133–141 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Gonçalves, D., Wong, P.: Homogeneous spaces in coincidence theory II. Forum Math., 17, 297–313 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Vendrúscolo, D., Wong, P.: Jiang-type theorems for coincidences of maps into homogeneous spaces. Topol. Methods Nonlinear Anal., 31, 151–160 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática-IME-Universidade de São PauloSão PauloBrazil
  2. 2.Department of MathematicsBates CollegeLewistonUSA

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