Smooth structures on sphere bundles over spheres (Q1113144)

In Bull. Am. Math. Soc. 75, 59-63 (1969; Zbl 0197.496), \textit{R. De Sapio} gave a classification of smooth structures of a p-sphere bundle over a q-sphere with one cross-section and \(p<q\). In Mich. Math. J. 14, 183-191 (1967; Zbl 0148.173), \textit{J. Munkres} also gave a classification up to concordance of differential structures in the case where the bundle has at least two cross-sections. In Invent. Math. 9, 81- 88 (1969; Zbl 0183.281), \textit{R. Schultz} gave a classification in the case \(p\geq q\). Here we will give a classification of the p-sphere bundle over a q-sphere without any cross-section and \(p<q\).