Line bundles with \(c_ 1(L)^ 2=0\). Higher order obstructions (Q1192783)

This paper is a continuation of a previous one [ibid. 2, No. 1, 83-90 (1991; Zbl 0728.57017)]. It concerns a further obstruction for a line bundle \(L\) with \(c_ 1(L)^ 2=0\) over a CW-complex \(M\) or an oriented \(n\)-dimensional manifold \(M\) to be the pull-back of a line bundle over \(S^ 2\). In the paper mentioned above the author found an obstruction in \(H^ 5(M,\mathbb{Z}_ 2)/Sq^ 2H^ 3\). Now he describes a further one in \(H^ 6(M,\mathbb{Z}_ 2)/Sq^ 2 H^ 4\) under some additional conditions. In the case of a smooth manifold \(M=M^ 6\) there is the following geometric description. Push the classifying map \(M\to\mathbb{C} P^ \infty\) of \(L\) to a smooth map \(M\to\mathbb{C} P^ 2\). The framed preimage of a generic point of \(\mathbb{C} P^ 2\) givs an element of the framed cobordism group \(\Omega^ 2_ k(pt)\cong\mathbb{Z}/2\) representing this new obstruction.