TY - JOUR T1 - Principal fibrations over noncommutative spheres JF - Reviews in Mathematical Physics Y1 - 2018 A1 - Michel Dubois-Violette A1 - Xiao Han A1 - Giovanni Landi AB - We present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes–Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres. VL - 30 UR - https://arxiv.org/abs/1804.07032 ER -