The chess conjecture (Q1408565)

Let \(P\) and \(Q\) be smooth orientable manifolds of dimensions \(p\) and \(q,\) respectively. Suppose that \(p\geq q\) and \(p-q\) is odd. The author proves that the homotopy class of an arbitrary Morin mapping \(f\colon P\rightarrow Q\) contains a cusp mapping. In fact, this implies the Chess conjecture [\textit{D. S. Chess}, Proc. Symp. Pure Math. 40, 221--224 (1983; Zbl 0523.58011)]. In addition, making use of results by [\textit{O. Saeki} and \textit{K. Sakuma}, Math. Proc. Camb. Phil. Soc. 124, 501--511 (1998; Zbl 0918.57009)] the author also obtains the following assertion. Suppose that Euler characteristic of \(P\) is odd and \(Q\) is an almost parallelizable manifold with odd dimension \(q\neq 1,3,7.\) Then there exist no Morin mappings from \(P\) into \(Q.\) In particular, such a manifold \(P\) does not admit Morin mappings into \(\mathbb R^{2k+1}\) for \(p \geq 2k+1 \neq 1,3,7\).