A fiberwise analogue of the Borsuk - Ulam theorem for sphere bundles over a 2-cell complex
Let \(\alpha\) be a finite dimensional real vector bundle over a CW-complex \(B\), and \(S(\alpha)\) be its sphere bundle with respect to some metric on \(\alpha\). The index of \(\alpha\), denoted by ind\,\(\alpha\), is defined to be the largest integer \(m\) for which there exists a \(\mathbb{Z}/2\)-map from \(S^{m-1}\) to \(S(\alpha)\), where \(S(\alpha)\) and \(S^{m-1}\) are regarded as \(\mathbb{Z}/2\)-spaces by the antipodal map (on each fibre). Let \(M^n(k)=S^{n-1}\cup_ke^n\) be the mapping cone of the degree \(k\) map \(k:S^{n-1}\to S^{n-1}\). In this paper, the following is proved: The equality ind\,\(\alpha=\dim\alpha\) holds for every vector bundle \(\alpha\) over the complex \(M^n(k)\), where \(n\geq 2\), if and only if (1) \(k\) is odd, (2) \(k=0\) and \(n\neq 2,3,4,5,8,9\), or (3) \(k\) is even, \(k>0\) and \(n\neq 2,3,4,8\).
scientific article; zbMATH DE number 5151668