Geodesic
Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds
Lee, Jong Bum1 ; Zhao, Xuezhi2
1Department of Mathematics, Sogang University, Seoul 121–742, KOREA
2Department of Mathematics, Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing 100037, CHINA
Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 563-580
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

For all continuous maps on 3–solvmanifolds, we give explicit formulas for a complete computation of the Nielsen type numbers NPn(f) and NΦn(f). The most general cases were explored by Heath and Keppelmann [Topology Appl. 76 (1997) 217–247] and the complementary part is studied in this paper. While studying the homotopy minimal periods of all maps on 3–solvmanifolds, we give a complete description of the sets of homotopy minimal periods of all such maps, including a correction to Jezierski, Kȩedra and Marzantowicz’s results in [Topology Appl. 144 (2004) 29–49].

DOI : 10.2140/agt.2008.8.563
Keywords: homotopy minimal period, Nielsen number, Nielsen type number, solvmanifold

Lee, Jong Bum  1   ; Zhao, Xuezhi  2

1 Department of Mathematics, Sogang University, Seoul 121–742, KOREA
2 Department of Mathematics, Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing 100037, CHINA 
@article{10_2140_agt_2008_8_563,
     author = {Lee, Jong Bum and Zhao, Xuezhi},
     title = {Nielsen type numbers and homotopy minimal periods for maps on 3{\textendash}solvmanifolds},
     journal = {Algebraic and Geometric Topology},
     pages = {563--580},
     year = {2008},
     volume = {8},
     number = {1},
     doi = {10.2140/agt.2008.8.563},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2008.8.563/}
}
TY  - JOUR
AU  - Lee, Jong Bum
AU  - Zhao, Xuezhi
TI  - Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds
JO  - Algebraic and Geometric Topology
PY  - 2008
SP  - 563
EP  - 580
VL  - 8
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2008.8.563/
DO  - 10.2140/agt.2008.8.563
ID  - 10_2140_agt_2008_8_563
ER  - 
%0 Journal Article
%A Lee, Jong Bum
%A Zhao, Xuezhi
%T Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds
%J Algebraic and Geometric Topology
%D 2008
%P 563-580
%V 8
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2008.8.563/
%R 10.2140/agt.2008.8.563
%F 10_2140_agt_2008_8_563
Lee, Jong Bum; Zhao, Xuezhi. Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds. Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 563-580. doi: 10.2140/agt.2008.8.563

[1] L Alsedà, S Baldwin, J Llibre, R Swanson, W Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math. 169 (1995) 1

[2] Y Ha, J Lee, Left invariant metrics and curvatures on simply connected three-dimensional Lie groups, to appear in Mathematische Nachrichten

[3] P R Heath, E Keppelmann, Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds. I, Topology Appl. 76 (1997) 217

[4] J Jezierski, Wecken's theorem for periodic points in dimension at least 3, Topology Appl. 153 (2006) 1825

[5] J Jezierski, J Kȩdra, W Marzantowicz, Homotopy minimal periods for NR-solvmanifolds maps, Topology Appl. 144 (2004) 29

[6] J Jezierski, W Marzantowicz, Homotopy minimal periods for maps of three-dimensional nilmanifolds, Pacific J. Math. 209 (2003) 85

[7] B Jiang, J Llibre, Minimal sets of periods for torus maps, Discrete Contin. Dynam. Systems 4 (1998) 301

[8] E C Keppelmann, C K Mccord, The Anosov theorem for exponential solvmanifolds, Pacific J. Math. 170 (1995) 143

[9] H J Kim, J B Lee, W S Yoo, Computation of the Nielsen type numbers for maps on the Klein bottle, to appear in J. Korean Math. Soc.

[10] J B Lee, X Zhao, Nielsen type numbers and homotopy minimal periods for all continuous maps on the $3$-nilmanifolds, Sci. China Ser. A 51 (2008) 351

[11] P Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983) 401

[12] W P Thurston, Three-dimensional geometry and topology. Vol. 1, Edited by Silvio Levy, Princeton Mathematical Series 35, Princeton University Press (1997)

Cité par Sources :