On embeddings of smooth covering spaces in vector bundles (Q1900739)

The author studies the existence of fibrewise embeddings of smooth coverings of \(n\)-dimensional manifolds in \(n\)-dimensional vector bundles. Such investigations were initiated by the reviewer in [Math. Ann. 236, 239-243 (1978; Zbl 0368.55001)]. The present paper contains generalizations and corrections to results of \textit{P. F. Duvall} and \textit{L. S. Husch} [Mem. Am. Math. Soc. 263 (1982; Zbl 0525.57012)]. In the main result, the author provides a fairly general criterion for the existence of a fibrewise embedding of a finite regular covering of a smooth closed \(n\)-manifold \(M\) into a smooth \(\mathbb{R}^n\)-bundle over \(M\) admitting a nonzero cross-section. The theorem is proved by calculating certain normal bordism groups inspired by work of Dax and Salomonsen from the early 1970's.