Topological Methods in Nonlinear Analysis

Nielsen fixed point theory on infra-solvmanifolds of Sol

Jang Hyun Jo and Jong Bum Lee

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Using averaging formulas, we compute the Lefschetz, Nielsen and Reidemeister numbers of maps on infra-solvmanifolds modeled on Sol, and we study the Jiang-type property for those infra-solvmanifolds.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 325-350.

Dates
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1491876032

Digital Object Identifier
doi:10.12775/TMNA.2016.080

Mathematical Reviews number (MathSciNet)
MR3635648

Zentralblatt MATH identifier
1382.55003

Citation

Jo, Jang Hyun; Lee, Jong Bum. Nielsen fixed point theory on infra-solvmanifolds of Sol. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 325--350. doi:10.12775/TMNA.2016.080. https://projecteuclid.org/euclid.tmna/1491876032


Export citation

References

  • A. Adler and J.E. Coury, The Theory of Numbers, Jones and Bartlett Publishers, Sudbury, 1995.
  • A. Fel'shtyn and J.B. Lee, The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type $\R$, Topology Appl. 181 (2015), 62–103.
  • D. Gonçalves and P. Wong, Nielsen numbers of selfmaps of $\Sol$ $3$-manifolds, Topology Appl. 159 (2012), 3729–3737.
  • K.Y. Ha and J.B. Lee, Crystallographic groups of $\Sol$, Math. Nachr. 286 (2013), 1614–1667.
  • ––––, The $R_\infty$ property for crystallographic groups of $\Sol$, Topology Appl. 181 (2015), 112–133.
  • ––––, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type $\R$ – Corrigendum, Nagoya Math. J. 221 (2016), 207–212.
  • K.Y. Ha, J.B. Lee and P. Penninckx, Anosov theorem for coincidences on special solvmanifolds of type $\R$, Proc. Amer. Math. Soc. 139 (2011), 2239–2248.
  • ––––, Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds, Fixed Point Theory Appl. 2012 (2012), 23 pp. \font\plsc=plcsc10 at 8pt
  • J. Jezierski, J. K\kedra and W. Marzantowicz, Homotopy minimal periods for $NR$-solvmanifolds maps, Topology Appl. 144 (2004), 29–49.
  • B. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983.
  • B. Jiang, S. Wang and Y.-Q. Wu, Homeomorphisms of $3$-manifolds and the realization of Nielsen number, Comm. Anal. Geom. 9 (2001), 825–878.
  • F.W. Kamber and Ph. Tondeur, Flat manifolds with parallel torsion, J. Differential Geometry 2 (1968), 385–389.
  • E.C. Keppelmann and C.K. McCord, The Anosov theorem for exponential solvmanifolds, Pacific J. Math. 170 (1995), 143–159.
  • S.W. Kim and J.B. Lee, Averaging formula for Nielsen coincidence numbers, Nagoya Math. J. 186 (2007), 69–93.
  • S.W. Kim, J.B. Lee and K.B. Lee, Averaging formula for Nielsen numbers, Nagoya Math. J. 178 (2005), 37–53.
  • J.B. Lee and K.B. Lee, Lefschetz numbers for continuous maps, and periods for expanding maps on infra-nilmanifolds, J. Geom. Phys. 56 (2006), 2011–2023.
  • ––––, Averaging formula for Nielsen numbers of maps on infra-solvmanifolds of type $\R$, Nagoya Math. J. 196 (2009), 117–134.
  • J.B. Lee and X. Zhao, Nielsen type numbers and homotopy minimal periods for maps on $3$-solvmanifolds, Alg. Geom. Topol. 8 (2008), 563–580.
  • W. Lück, Survey on aspherical manifolds, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, 53–82.
  • F. Mangolte and J.-Y. Welschinger, Do uniruled six-manifolds contain $\Sol$ Lagrangian submanifolds?, Int. Math. Res. Not. IMRN 63 (2011), 1–34.
  • H. Sun, S. Wang and J. Wu, Self-mapping degrees of torus bundles and torus semi-bundles, Osaka J. Math. 47 (2010), 131–155.
  • F. Wecken, Fixpunktklassen. \romIII. Mindestzahlen von Fixpunkten, Math. Ann. 118 (1942), 544–577.
  • B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2000), 195–237.
  • P. Wong, Fixed-point theory for homogeneous spaces, Amer. J. Math. 120 (1998), 23–42.