Cobordism classes of fiber bundles (Q5905411)

Let \({\mathcal R}_ n\) be an unoriented cobordism group of smooth closed \(n\)-dimensional manifolds and \({\mathcal R}_ \bullet = \sum_ n {\mathcal R}_ n\) be a cobordism ring. Let \([M]_ 2\) be the class of the manifold \(M^ n\) in \({\mathcal R}_ n\). For a given manifold \(N^ m\), we consider the classes \(\alpha \in {\mathcal R}_ n\). Is there a fiber bundle over \(N^ m\) whose total space is a smooth closed manifold \(M^ n\) such that \([M]_ 2 = \alpha\)? In the cases \(N^ m = S^ 1\), \(S^ 2\), RP(2), this problem has been solved [\textit{R. E. Stong}, Trans. Am. Math. Soc. 178, 431-447 (1973; Zbl 0267.57025)]. We shall discuss \(N^ m = \text{RP} (3)\), \((S^ 1)^ 2 \times S^ 2\) and \(S^ 2 \times S^ 2\). Our main results are the following. Theorem 1: A class \(\alpha \in {\mathcal R}_ n\) is represented by a fiber bundle over RP(3) iff its Stiefel- Whitney numbers involving \(W_ n\), \(W_{n-1}\) and \(W_{n-2}\) are all zero. Theorem 2: The set of classes represented by fiber bundles over \((S^ 1)^ 2 \times S^ 2\) is precisely the set of classes in which the Stiefel-Whitney numbers divisible by \(W_ n\), \(W_{n-1}\), \(W_{n- 2}\) and \(W_{n-3}\) are zero. In addition, we give a sufficient condition for the classes to be represented by fiber bundles over \(S^ i\) \((i = 1,2,4,8)\).