  We extend Lawrence's representations of the braid groups to relative homology
modules, and we show that they are free modules over a Laurent polynomials
ring. We define homological operators and we show that they actually provide a
representation for an integral version for $U_q \mathfrak{sl}(2)$. We suggest
an isomorphism between a given basis of homological modules and the standard
basis of tensor products of Verma modules, and we show it to preserve the
integral ring of coefficients, the action of $U_q \mathfrak{sl}(2)$, the braid
group representations and their grading. This recovers an integral version for
Kohno's theorem relating absolute Lawrence representations with quantum braid
representation on highest weight vectors. It is an extension of the latter
theorem as we get rid of generic conditions on parameters, and as we recover
the entire product of Verma-modules as a braid group and a $U_q
\mathfrak{sl}(2)$-module.
