Nielsen fixed point theory on infra-solvmanifolds of $\mathrm{Sol}$

Jang Hyun Jo, Jong Bum Lee

DOI: http://dx.doi.org/10.12775/TMNA.2016.080

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.

Keywords


Averaging formula; infra-solvmanifold; Jiang-type space; Lefschetz number; Nielsen number; Reidemeister number; Sol-geometry

Full Text:

PREVIEW FULL TEXT

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 infrasolvmanifolds 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.

K.Y. Ha and J.B. Lee, The R∞ property for crystallographic groups of Sol, Topology Appl. 181 (2015), 112–133.

K.Y. Ha and J.B. Lee, 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.

K.Y. Ha, J.B. Lee and P. Penninckx, Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds, Fixed Point Theory Appl. 2012 (2012), 23 pp.

J. Jezierski, J. Kędra and W. Marzantowicz, Homotopy minimal periods for N Rsolvmanifolds 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.

J.B. Lee and K.B. Lee, 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 semibundles, Osaka J. Math. 47 (2010), 131–155.

F. Wecken, Fixpunktklassen. III. 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.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism