Algebraic and Geometric Topology 3 (2003), paper no. 26, pages 777-789.

The Chess conjecture

Rustam Sadykov

Abstract. We prove that the homotopy class of a Morin mapping f: P^p --> Q^q 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 [S_1^k(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 P^p with odd Euler characteristic does not admit Morin mappings into R^{2k+1} for p > 2k not equal to 1,3 or 7.

Keywords. Singularities, cusps, fold mappings, jets

AMS subject classification. Primary: 57R45. Secondary: 58A20, 58K30.

DOI: 10.2140/agt.2003.3.777

E-print: arXiv:math.GT/0301371

Submitted: 18 February 2003. (Revised: 23 July 2003.) Accepted: 19 August 2003. Published: 24 August 2003.

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Rustam Sadykov
University of Florida, Department of Mathematics
358 Little Hall, 118105, Gainesville, Fl, 32611-8105, USA

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