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The chess conjecture. (English) Zbl 1036.57012

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 [D. S. Chess, Proc. Symp. Pure Math. 40, 221–224 (1983; Zbl 0523.58011)]. In addition, making use of results by [O. Saeki and 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 \(p \geq 2k+1 \neq 1,3,7\).

MSC:

57R45 Singularities of differentiable mappings in differential topology
58A20 Jets in global analysis
58K30 Global theory of singularities
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References:

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