3-sphere fibrations: a tool for analyzing twisted materials in condensed matter

Chiral molecules, when densely packed in soft condensed matter or biological materials, build organizations which are most often spontaneously twisted. The crystals of 'blue' phases formed by small mesogenic molecules demonstrate the structural importance of such a twist or torsion, and its presence was also recently observed in finite toroidal aggregates formed by long DNA molecules. The formation of these organizations is driven by the fact that compactness, which tends to align the molecules, enters into conflict with torsion, which tends to disrupt this alignment. This conflict of topological nature, or frustration, arises because of the flatness of the Euclidean space, but does not exist in the curved space of the 3-sphere where particular lines, its fibres, can be drawn which are parallel and nevertheless twisted. As these fibrations conciliate compactness and torsion, they can be used as geometrical templates for the analysis of organizations in the Euclidean space. We describe these fibrations, with a particular emphasis on their torsion.