On the vector field problem for product manifolds
\(\text{Span}(M)\) denotes the maximal number of everywhere linearly independent tangent vector fields on a smooth compact manifold \(M\). Using K-theory methods the authors prove a criterion which gives an upper bound for \(\text{Span}(M)\), when \(M\) is a product of two stably complex manifolds. The criterion is applied to the product of lens spaces \(L^{n_1}(2^{m_1}) \times L^{n_2}(2^{m_2})\) and to the product \(N^{n_1}(m_1) \times N^{n_2}(m_2)\) of quaternionic spherical space forms, when the numbers \(m_1\) and \(m_2\) are large enough, and it is applied to products of Dold manifolds. In the case of lens spaces these upper bounds are equal to Span by a result of \textit{J. C. Becker} [Am. J. Math. 94, 991-1026 (1972; Zbl 0258.57005)].
scientific article; zbMATH DE number 1251458