The order of real line bundles (Q1885517)

Let \(\xi\) be a real line bundle over a space \(X\) having the homotopy type of a finite CW-complex. For any \(1 \leq r \leq n\), set \(n-r=s\), \(c=[s/2]\), \(n=2m\) or \(n=2m+1\) and also put \[ a_0(n, r)= \begin{cases} \min\{2j-1+\nu_2{m \choose j}:c+1\leq j \} & \text{if \(nr\) is even} \\\min\{2c+\nu_2{m \choose c}, 2j-1+\nu_2{m \choose j}:c+1\leq j \} & \text{otherwise.} \end{cases} \] Then the main result of the paper asserts that if \(n\xi\) admits \(r\geq 1\) independent sections, then \(2^{a_0(n, r)}\xi\) is stably trivial. This result contributes towards sharpening some previous results in this direction. In fact, the following application shows that this upper bound is the best possible. Let \(X_{n, r}\) be the projective Stiefel manifold which is defined to be a double covering space over the Stiefel manifold \(V_{n, r}\). Denote by \(\xi_{n, r}\) the canonical line bundle associated to this double covering. Then one finds that \(n\xi_{n, r}\) admits \(r\) independent sections. This fact enables one to determine the order of \(\xi_{n, r}\) and consequently one has that if either \(n\equiv 0, \pm 1 \bmod 8\) or \(a_0(n, r) \leq [(n-1)/2]\), then the order of \(\xi_{n, r}\) equals \(2^{a(n, r)}\) where \(a(n, r)=\min \{ [(n-1)/2], a_0(n, r)\}\). The authors pose a conjecture for the case where \(n \not\equiv 0, \pm 1 \mod 8\) and \(a_0(n, r) > [(n-1)/2]\). Furthermore, various applications of these two theorems are given in the final section.