Characteristic classes of vector bundles over \( CP(j)\times HP(k)\) and involutions fixing \( CP(2m+1)\times HP(k)\) (Q2897920)

Let \(F\) be a closed manifold and \(M\) be a smooth and closed manifold equipped with a smooth involution \(T:M \to M\) whose fixed point set is \(F\). When \(F\) is the disjoint (finite) union of smooth and closed manifolds there are many results in the literature, but there are few results [\textit{P. L. Q. Pergher}, Manuscr. Math. 89, No. 4, 471--474 (1996; Zbl 0860.57030)] and [\textit{R. E. Stong}, Proc. Am. Math. Soc. 119, No. 3, 1005--1008 (1993; Zbl 0810.57023)] when considering a product of spaces.NEWLINENEWLINEIn this work, the authors consider the case where \(F= \mathbb{C}P(2m+1) \times \mathbb{H}P(k)\) and show that every involution fixing the set \(F\) bounds. For the proof of this result they determine the total Stiefel-Whitney classes of vector bundles over \(\mathbb{C}P(j) \times \mathbb{H}P(k).\)