On unit sphere tangent bundles over complex Grassmannians (Q6558200)

A simply connected space \(X\) is \textit{formal} if there exists a quasi-isomorphism \((\wedge V,d)\rightarrow H^*(\wedge V,d)\), where \((\wedge V,d)\) is the minimal Sullivan model for \(X\). For example, it is known that every compact Kähler manifold is formal. In particular, complex projective spaces, and more generally, complex Grassmannians, are formal. The known examples of formal spaces also include spheres of arbitrary dimensions.\N\NThe main result of the present paper states that the total space \(E\) of the sphere tangent bundle \(S^{2k(n-k)-1}\rightarrow E\rightarrow G_k(\mathbb C^n)\) over the complex Grassmann manifold \(G_k(\mathbb C^n)\) is not formal (although both the fiber and the base space are). The authors use a previously known fact that all triple Massey products in formal spaces are trivial, and thus prove their result by detecting a nontrivial triple Massey product in \(E\).