The Chess conjecture

We prove that the homotopy class of a Morin mapping $f : P^{p} \to Q^{q}$ 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 $ℝ^{2 k + 1}$ for $p \geq 2 k + 1 \neq 1, 3, 7$ .

Keywords

singularities, cusps, fold mappings, jets

Mathematical Subject Classification 2000

Primary: 57R45

Secondary: 58A20, 58K30

References

Forward citations

Publication

Received: 18 February 2003

Revised: 23 July 2003

Accepted: 19 August 2003

Published: 24 August 2003

Authors

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

Volume 3, issue 2 (2003)

Rustam Sadykov