Vector fields on projective Stiefel manifolds and the Browder-Dupont invariant (Q2216591)

The projective Stiefel manifold \(X_{n,r}\) is the quotient of the Stiefel manifold \(V_{n,r}\) of orthogonal \(r\)-frames in \(\mathbb R^n\) by the \(\mathbb Z_2\)-action given by \((v_1,\ldots,v_r)\mapsto(-v_1,\ldots,-v_r)\). This paper studies the span of \(X_{n,r}\). The span of a vector bundle is the maximal number of everywhere linearly independent sections of the bundle, and the span of a smooth manifold is the span of its tangent bundle, i.e., it is the maximal number of linearly independent vector fields on the manifold. The stable span of a manifold is defined by \[\mathrm{span}^0(M):=\mathrm{span}(\tau_M\oplus t\varepsilon)-t,\] where \(\tau_M\) is the tangent bundle of \(M\), \(\varepsilon\) is the trivial line bundle on \(M\), and \(t\) is any positive integer. It is obvious that \(\mathrm{span}^0(M)\geq\mathrm{span}(M)\). The authors of the paper exhibit a lower bound for \(\mathrm{span}^0(X_{n,r})\), which is denoted by \(k_{n,r}\), and is given in terms of the span of multiples of the Hopf bundle over real projective space \(P^{n-1}\) (which is extensively studied and computable). One of the main results of the paper gives a number of cases in which \(k_{n,r}\) is a lower bound for \(\mathrm{span}(X_{n,r})\) as well. More precisely, \(\mathrm{span}(X_{n,r})\geq k_{n,r}\) if any of the following three conditions hold: \begin{itemize} \item \(r>2\), \item \(r=2\) and \(n\) is even, \item \(r=2\), \(n=3\) or \(n\) is odd with \(\chi(n)=0\); \end{itemize} where \(\chi(n)\in\mathbb Z_2\) is a mod \(2\) constant. The consideration of the third case (\(r=2\), \(n\) odd) occupies the major part of the proof and uses the Browder-Dupont invariant. The authors also use Stiefel-Whitney classes to find upper bounds for \(\mathrm{span}(X_{n,r})\) in some cases. Some of the results thus obtained are: \(\mathrm{span}(X_{5,2})=5\), \(\mathrm{span}(X_{13,2})=13\), \(\mathrm{span}(X_{16,5})=58\), \(1618\leq\mathrm{span}(X_{58,51})\leq1625\).