00970nas a2200145 4500008004100000245005300041210005300094300001200147490000700159520055900166100002800725700001400753700002000767856003700787 2018 eng d00aPrincipal fibrations over noncommutative spheres0 aPrincipal fibrations over noncommutative spheres a18500200 v303 aWe 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 Connesâ€“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.1 aDubois-Violette, Michel1 aHan, Xiao1 aLandi, Giovanni uhttps://arxiv.org/abs/1804.07032