A study of topology of the flip Stiefel manifolds (Q6606292)

The Stiefel manifold \(V_{n,k}\) is defined as the set of orthogonal \(k\)-frames in \(\mathbb R^n\). A well-known action of the cyclic group \(\mathbb Z/2\mathbb Z\) on \(V_{n,k}\) is given by the action of the generator: \((v_1,v_2,\ldots,v_k)\mapsto(-v_1,-v_2,\ldots,-v_k)\). The obtained orbit space is called the projective Stiefel manifold and it is denoted by \(PV_{n,k}\). The mod \(2\) cohomology algebra of \(PV_{n,k}\) was calculated in [\textit{S. Gitler} and \textit{D. Handel}, Topology 7, 39--46 (1968; Zbl 0166.19405)].\N\NThe present paper considers another free action of \(\mathbb Z/2\mathbb Z\) on \(V_{n,k}\) in the case when \(k\) is even; namely, the generator of \(\mathbb Z/2\mathbb Z\) acts by simultaneous pairwise flipping of the coordinates:\N\[\N(v_1,v_2,v_3,v_4,\ldots,v_{k-1},v_k)\mapsto(v_2,v_1,v_4,v_3,\ldots,v_k,v_{k-1}).\N\]\NThe orbit space of this action is denoted by \(FV_{n,k}\) and is called the \textit{flip Stiefel manifold}. The authors compute the additive structure of the mod \(2\) cohomology of \(FV_{n,k}\) by inspecting the Serre spectral sequence of the fibration \(V_{n,k}\rightarrow FV_{n,k}\rightarrow B(\mathbb Z/2\mathbb Z)=\mathbb R\mathrm P^\infty\). Then they use these results on cohomology, along with some properties of the tangent bundle over \(FV_{n,k}\), to calculate the total Stiefel-Whitney class of this manifold. Since the parallelizability of a manifold implies the vanishing of its Stiefel-Whitney classes, the previous results are used to detect a few infinite families of non-paralellizable flip Stiefel manifolds. Some sufficient conditions for the equality of the span and the stable span of \(FV_{n,k}\) are also given.\N\NThe paper is concluded with a discussion on the existence of equivariant maps between some Stiefel manifolds (involving both mentioned actions of \(\mathbb Z/2\mathbb Z\)), and some theorems of Borsuk-Ulam type are obtained. For example, the authors apply their results to prove that for any (continuous) map \(f:S^n\rightarrow\mathbb R^{n-1}\) there exist \(v_1,v_2\in S^n\) such that \(v_1\perp v_2\) and \(f(v_1)=f(v_2)\).