Abstract
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.
Similar content being viewed by others
References
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)
Brooks, R. B. S.: On the sharpness of the Δ2 and Δ1 Nielsen numbers. J. Reine Angew. Math., 259, 101–108 (1973)
Gonçalves, D., Jezierski, J., Wong, P.: Obstruction theory and coincidences in positive codimension. Acta Math. Sin. Engl. Ser., 22(5), 1591–1602 (2006)
Gonçalves, D., Wong, P.: Wecken property for roots. Proc. Amer. Math. Soc., 133(9), 2779–2782 (2005)
Gonçalves, D., Wong, P.: Obstruction theory and coincidences of maps between nilmanifolds. Archiv Math., 84, 568–576 (2005)
Gonçalves, D., Wong, P.: Nilmanifolds are Jiang-type spaces for coincidences. Forum Math., 13, 133–141 (2001)
Gonçalves, D., Wong, P.: Homogeneous spaces in coincidence theory II. Forum Math., 17, 297–313 (2005)
Vendrúscolo, D., Wong, P.: Jiang-type theorems for coincidences of maps into homogeneous spaces. Topol. Methods Nonlinear Anal., 31, 151–160 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported in part by Projeto Temático Topologia Algébrica Geométrica e Differencial (Grant No. 2016/24707-4)
Rights and permissions
About this article
Cite this article
Gonçalves, D., Wong, P. Coincidence Wecken Property for Nilmanifolds. Acta. Math. Sin.-English Ser. 35, 239–244 (2019). https://doi.org/10.1007/s10114-018-7315-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-018-7315-3