The Chess conjecture

Abstract

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 $\left[S_1^k\left(f\right)\right]$, 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 $p\ge 2k+1\ne 1,3,7$.