On the embeddability of skeleta of spheres (Q2655789)

Generalizations of the classical van Kampen-Flores theorem are discussed. The main result states the following. Theorem 1. Let \(S_{\leq d}\) be the \(d\)-skeleton of a piecewise linear \(2d\)-sphere and let \(M\) be a missing \(d\)-face of \(S_{\leq d}\), i.e. \(\dim M=d\), \(M\not\in S_{\leq d}\). Let the boundary \(\partial M\) be contained in \(S_{\leq d}\). Then the union \(K=S_{\leq d}\cup M\) does not embed in the \(2d\)-sphere. Here piecewise linearity means that the sphere is obtained by finitely many bistellar moves (Pachner moves) from the boundary of a simplex. It is conjectured that the condition of piecewise linearity is superfluous. Another generalization deals with triangulated spheres. Theorem 2. Let \(K\) be the \(d\)-skeleton of a triangulated (\(2d+1\))-sphere. Then \(K\) does not embed in the \(2d\)-sphere. It is conjectured that \(K\) as in Theorem 2 does not contain a subdivision of the \(d\)-skeleton of the (\(2d+2\))-sphere, if \(d>1\). The obtained results are related to the \(g\)-conjecture for simplicial spheres, cf. [\textit{P. McMullen}, Isr. J. Math. 9, 559--570 (1971; Zbl 0209.53701); \textit{G. Kalai}, Adv. Stud. Pure Math. 33, 121--163 (2001; Zbl 1034.57021)].