Normal characteristic numbers (Q2781313)

Let \(M^{4n}\) be a closed oriented smooth manifold of dimension \(4n\), and let \(\overline p_n[M^{4n}]\) denote the \(n\)th Pontryagin class of the stable normal bundle of \(M\) evaluated on the fundamental homology class of \(M\). It is proved that the greatest common divisor of the integers \(\overline p_n[M^{4n}]\) is \(3^k\), where \(k\) is the smallest integer \(j\) for which \(\alpha_3 (2n+j) \leq 3j\). For any integer \(x\) the number \(\alpha_3 (x)\) is the sum of the digits in the 3-adic expansion of \(x\). The integers \(\overline p_n [M^{4n}]\) are of interest in connection with the work of \textit{A. Szücs} [On the singularities of hyperplane projections of immersions, Bull. Lond. Math. Soc. 32, 364-374 (2000; Zbl 1021.57013)]. A similar result is proved for the common divisor of the normal Chern numbers \(\overline c_n[M^{2n}]\) of stably complex manifolds of dimension \(2n\).