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On tangent bundles of certain homogeneous spaces. (English) Zbl 1062.57031

Let be \(X_n=SO(n)/(SO(n-4)\times SU(2))\) and \(Y_n=SO(n)/(SO(n-4)\times U(2))\), \(n \geq 4\). The authors compute the total Chern class \(c(Y_n)\) of the tangent bundle of \(Y_n\), and then the total Stiefel-Whitney class \(w(X_n)\) and the total Pontrjagin class \(p(X_n)\) of the tangent bundle of \(X_n\). In particular they prove that \(X_n\) is parallelizable if and only if \(n=4\) or \(5\).

MSC:

57R20 Characteristic classes and numbers in differential topology
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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