Lifting obstructions, ordinary obstructions and spherical fibrations (Q1347779)

Suppose that \(S^3 \hookrightarrow S^7 \to S^4\) is the standard \(S^3\)-bundle over the quaternionic projective line \(S^4\), \(M\) is a smooth closed oriented 8-manifold, and \(f: M \to S^4\) is a map. The author studies obstructions to the lifting of \(f\). In [\textit{C. Bohr}, Topology Appl. 103, No. 3, 283-290 (2000; Zbl 0979.55012)] the author studied such a question for the Hopf bundle \(S^1 \hookrightarrow S^3 \to S^2\), but the quaternionic case, considered in the paper under review, requires new methods. It is always possible to find a lifting \(\overline {f}\) of \(f\) outside a smooth submanifold of \(M\) of codimension 4. Such a lifting \(\overline {f} : M-\Delta \to S^7\) , where \(\Delta \subset M\) is a smooth closed submanifold of codimension 4, is referred to as a ``defect lifting'', and \(\Delta\) as the ``defect set'' of \(\overline {f}\); this is a terminology used in physics. For each connected component \(\Delta_i\) of \(\Delta\), a local index of \(\overline {f}\) can be defined; it can be seen as the first obstruction to extending \(\overline {f}\) over a small disk normal to \(\Delta_i\). Next, by looking at tubular neighborhoods of the components \(\Delta_i\), obstruction to extending the lifting as an ordinary map to \(S^7\) can be defined. The main result of the paper establishes a relationship between the two kinds of obstruction. Applications and examples illustrating the main results are given.