The local defect index up to finite ambiguity
Let \(E\) be a \(q\)-dimensional oriented vector bundle with \(q\geq 3\) over a \((p+1)\)-dimensional cell complex \(X\) and let \(\sigma\) be a nowhere vanishing cross section over the \(p\)-skeleton \(X_p\) of \(X\). Then we have a well-known cohomology class \(c(E,\sigma)\) in \(H^{p+1} (X;\pi_p(S^{q-1}))\) which is the primary obstruction to extending \(\sigma|X_{p-1}\) to \(X\). The authors consider especially the rational class of this obstruction. However we know that the coefficient group is infinite only if \(p=q-1\) or \(p=4\ell-1\), \(q=2\ell+1\) and further in the first case \(c(E,\sigma)\) is just the Euler class of \(E\). So the present paper deals with the second case. The result is \(p_\ell(E)-e( \sigma)^2= (-1)^\ell 4h_*(c(E,\sigma)) \in H^{4\ell}(X:\mathbb{Q})\) where \(p_\ell (E)\) denotes the \(\ell\)th Pontryagin class of \(E\), \(e(\sigma)\) the Euler class of the complementary bundle of \(\sigma\) and \(h:\pi_{4\ell-1}(S^{2\ell}) \to\mathbb{Z} \subset\mathbb{Q}\) the Hopf invariant.
scientific article; zbMATH DE number 1741952