The power of the normal bundle associated to an immersion of \(RP^n\), its complexification and extendibility (Q2643504)

Let \(\nu\) be the normal bundle associated to an immersion of the \(n\)-dimensional real projective space \(\mathbb RP^n\) in \(\mathbb{R}^{n+k}\). The authors consider the problem of extendibility of \(\nu^r=\nu\otimes\cdots\otimes\nu\) (\(r\)-fold) to \(\mathbb RP^m\) for \(m \geq n\). Now a \(t\)-dimensional real vector bundle \(\zeta\) over \(\mathbb RP^n\) is said to be extendible (resp. stably extendible) to \(\mathbb RP^m\) for \(m \geq n\) if there exists a \(t\)-dimensional real vector bundle \(\alpha\) over \(\mathbb RP^m\) such that \(\alpha| \mathbb RP^n\) is equivalent (resp. stably equivalent) to \(\zeta\). The main result of this paper contains three theorems. The first one (Theorem A) proves that, assuming that \(n < k^r\), the following are equivalent: (1) \(\nu^r\) is extendible to \(\mathbb RP^m\) for every \(m \geq n\). (2) \(\nu^r\) is stably extendible to \(\mathbb RP^m\) for every \(m \geq n\). (3) There is an integer \(a\) satisfying \((2n+k+2)^r-k^r \leq a2^{\phi(n)+1} \leq (2n+k+2)^r+k^r\), where \(\phi(n)\) denotes the number of integers \(s\) such that \(0 < s \leq n\) and \(s \equiv 0, 1, 2\) or \(4 \bmod 8\). This is a generalization of the results on \(\nu\) and \(\nu^2\) obtained by the second author, \textit{H. Maki} and \textit{T. Yoshida} [Osaka J. Math. 39, 315--324 (2002; Zbl 1007.55012)] and by the second author and \textit{K. Komatsu} [Hiroshima Math. J. 32, 371--378 (2002; Zbl 1020.55002)]. The first paper quoted above now contributes greatly to the proofs of this theorem and the next one. In fact the authors use two theorems established there. The second one (Theorem B) shows that a similar result holds for the complexification \(c\nu^r\) of \(\nu^r\). Its explicit statement can be obtained by replacing, in the above statement, \(\nu^r\) by \(c\nu^r\) and two inequalities by \(\langle n/2 \rangle \leq k^r\) and \((2n+k+2)^r-k^r \leq a2^{[n/2]+1} \leq (2n+k+2)^r+k^r\), respectively, where \(\langle x \rangle\) denotes the smallest integer \(n\) with \(x \leq n\). But it happens that these two theorems deal only with the case where the two kinds of notions of extendibility, that is, extendibility and stable extendibility, are equivalent. The last one (Theorem C) provides an example which clarifies the distinction between them. It states that there is a 2-dimensional real vector bundle over \(\mathbb RP^2\) which is stably extendible to \(\mathbb RP^3\) but is not extendible to \(\mathbb RP^3\).