Immersions and embeddings of orientable manifolds up to unoriented cobordism
Let \(\alpha(n)\) denote the number of ones in the binary expansion of \(n\). \textit{R. L. W. Brown} [Can. J. Math. 23, 1102-1115 (1971; Zbl 0216.20104)] has proved that every differentiable closed \(n\)-manifold \(M^n\) is cobordant to a manifold that immerses in \(R^{2n-\alpha (n)}\) and embeds into \(R^{2n- \alpha(n)+1}\). If some additional conditions are placed on \(M^n\), then it is possible to lower the dimension to \(2n-\alpha (n)\), respectively, \(2n-\alpha (n) +1\). Some such conditions can be already found in Brown's cited paper. The author now places the orientability of \(M^n\) as the additional condition in Brown's theorem. Writing \(n\) in the form \(n= (2m+1) 2^{\nu(n)}\), let \(\beta(n) =2n- \alpha(n)- \min \{\alpha(n), \nu(n)\}\). Then as the main result one could consider the following statement: Any orientable manifold \(M^n\) immerses (embeds) up to unoriented cobordism in \(R^{\beta(n)}\) \((R^{\beta (n)+1})\). Also, depending on certain properties of \(n\), \(\beta(n)\) and \(\nu(n)\) it is shown that in such cases \(M^n\) immerses (embeds) up to unoriented cobordism in \(R^{\beta (n)-1}\) \((R^{\beta (n)})\).
scientific article; zbMATH DE number 1021653