#### Volume 3, issue 2 (2003)

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The Chess conjecture

Algebraic & Geometric Topology 3 (2003) 777–789
 arXiv: math.GT/0301371
##### Abstract

We prove that the homotopy class of a Morin mapping $f:\phantom{\rule{0.3em}{0ex}}{P}^{p}\to {Q}^{q}$ 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 $p\ge 2k+1\ne 1,3,7$.

##### Keywords
singularities, cusps, fold mappings, jets
##### Mathematical Subject Classification 2000
Primary: 57R45
Secondary: 58A20, 58K30