Geodesic
The Chess conjecture
Sadykov, Rustam1
1University of Florida, Department of Mathematics, 358 Little Hall, 118105, Gainesville, FL 32611-8105, USA
Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 777-789
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We prove that the homotopy class of a Morin mapping f : Pp → Qq with p − q odd contains a cusp mapping. This affirmatively solves a strengthened version of the Chess conjecture [DS Chess, A note on the classes [S1k(f)], Proc. Symp. Pure Math., 40 (1983) 221–224] and [VI Arnol’d, VA Vasil’ev, VV Goryunov, OV Lyashenko, Dynamical systems VI. Singularities, local and global theory, Encyclopedia of Mathematical Sciences - Vol. 6 (Springer, Berlin, 1993)]. Also, in view of the Saeki–Sakuma theorem [O Saeki, K Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, Math. Proc. Camb. Phil. Soc. 124 (1998) 501–511] on the Hopf invariant one problem and Morin mappings, this implies that a manifold Pp with odd Euler characteristic does not admit Morin mappings into ℝ2k+1 for p ≥ 2k + 1≠1,3,7.

DOI : 10.2140/agt.2003.3.777
Keywords: singularities, cusps, fold mappings, jets

Sadykov, Rustam  1

1 University of Florida, Department of Mathematics, 358 Little Hall, 118105, Gainesville, FL 32611-8105, USA 
@article{10_2140_agt_2003_3_777,
     author = {Sadykov, Rustam},
     title = {The {Chess} conjecture},
     journal = {Algebraic and Geometric Topology},
     pages = {777--789},
     year = {2003},
     volume = {3},
     number = {2},
     doi = {10.2140/agt.2003.3.777},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2003.3.777/}
}
TY  - JOUR
AU  - Sadykov, Rustam
TI  - The Chess conjecture
JO  - Algebraic and Geometric Topology
PY  - 2003
SP  - 777
EP  - 789
VL  - 3
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2003.3.777/
DO  - 10.2140/agt.2003.3.777
ID  - 10_2140_agt_2003_3_777
ER  - 
%0 Journal Article
%A Sadykov, Rustam
%T The Chess conjecture
%J Algebraic and Geometric Topology
%D 2003
%P 777-789
%V 3
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2003.3.777/
%R 10.2140/agt.2003.3.777
%F 10_2140_agt_2003_3_777
Sadykov, Rustam. The Chess conjecture. Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 777-789. doi: 10.2140/agt.2003.3.777

[1] P M Akhmetiev, R R Sadykov, A remark on elimination of singularities for mappings of 4–manifolds into 3–manifolds, Topology Appl. 131 (2003) 51

[2] Y Ando, On the elimination of Morin singularities, J. Math. Soc. Japan 37 (1985) 471

[3] V I Arnol’D, V A Vasil’Ev, V V Goryunov, O V Lyashenko, Dynamical systems VI: Singularities, local and global theory, Encyclopaedia of Mathematical Sciences 6, Springer (1993)

[4] J M Boardman, Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. (1967) 21

[5] D S Chess, A note on the classes $[S^{k}_{1}(f)]$, from: "Singularities, Part 1 (Arcata, Calif., 1981)", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 221

[6] Y Eliashberg, Surgery of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 1321

[7] S Kikuchi, O Saeki, Remarks on the topology of folds, Proc. Amer. Math. Soc. 123 (1995) 905

[8] F Ronga, Le calcul des classes duales aux singularités de Boardman d'ordre deux, Comment. Math. Helv. 47 (1972) 15

[9] O Saeki, Notes on the topology of folds, J. Math. Soc. Japan 44 (1992) 551

[10] O Saeki, K Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, Math. Proc. Cambridge Philos. Soc. 124 (1998) 501

[11] O Saeki, K Sakuma, Elimination of singularities: Thom polynomials and beyond, from: "Singularity theory (Liverpool, 1996)", London Math. Soc. Lecture Note Ser. 263, Cambridge Univ. Press (1999) 291

[12] K Sakuma, A note on nonremovable cusp singularities, Hiroshima Math. J. 31 (2001) 461

Cité par Sources :