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The vector field problem for projective Stiefel manifolds. (English) Zbl 1222.57026

The projective Stiefel manifolds \(X_{n,r}, 1 \leq r <n\) are obtained from the Stiefel manifolds \(V_{n,r}\) of orthonormal \(r\)-frames in \(\mathbb{R}^n\) by identifying \((v_1, \cdots , v_r) \in V_{n,r}\) with \((-v_1, \cdots , -v_r)\). In this paper the problem of determining the span of the projective Stiefel manifolds \(X_{n,r}, r \geq 2\), is extensively considered. Besides some new results, the authors present an overview on the subject which naturally leads them to conjecture further results. By doing this, they show the wide range of techniques the problem involves. For instance, the case \(r=2\) and \( n\) odd presents additional dificulties and then the Browder-Dupont invariant is brought in. For \( n = 8m -1\), an improvement for a lower bound for \(\text{span}(X_{n,2})\) is obtained by using Hurwitz-Radon multiplications: one has \(\text{span}(X_{n,2}) \geq \text{span}(S^n)\). Also the Hurwitz-Radon multiplications are carried out here in an explicit version.

MSC:

57R25 Vector fields, frame fields in differential topology
57R19 Algebraic topology on manifolds and differential topology
57R20 Characteristic classes and numbers in differential topology
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