Celebratio Mathematica — Whitney

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Works connected to Albrecht Dold

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A. Dold and H. Whitney: “Classification of oriented sphere bundles over a 4-complex,” Ann. Math. (2) 69 : 3 (May 1959), pp. 667–677. MR 0123331 Zbl 0124.38103 article

The classification of \( (n-1) \)-sphere bundles with structure group \( SO(n) \) (special orthogonal group) over a complex \( K \) of dimension at most 4 has been carried out in several special cases. If \( n=2 \) or if the dimension of \( K \) does not exceed 3 then the characteristic class \( W_2 \) is a complete invariant (see [1937]; note that \( W_2 \) is an integer class for \( n=2 \), and is a class mod 2 if \( n\geq 3 \)). 2-sphere bundles with vanishing class \( W_2 \) were classified by the second author in an unpublishedanuscript (1938; announced in [1940]); these bundles are not determined by their characteristic classes (see our example in Section 3). Pontrjagin in [1945] gave a solution for arbitrary \( n \) provided that \( H^4(K;Z) \) has no 2-torsion; in this case \( (n-1) \)-sphere bundles are characterized by \( W_2 \), \( W_4 \) (for \( n\geq 4 \)) and \( P_4 \) (see [Pontrjagin 1945] or the Corollary in Section 3). In this paper we give the classification for the general case. Throughout the paper we assume \( n\geq 3 \).