In this talk, we consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We present a sharp lower bound on the jump length of the lifting, established in a recent joint work with G. Canevari (Università di Verona) and B. Stroffolini (Università degli studi di Napoli Federico II), and expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.