%0 Journal Article %J Reviews in Mathematical Physics %D 2018 %T Principal fibrations over noncommutative spheres %A Michel Dubois-Violette %A Xiao Han %A Giovanni Landi %X 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. %B Reviews in Mathematical Physics %V 30 %P 1850020 %G eng %U https://arxiv.org/abs/1804.07032 %R 10.1142/S0129055X18500204