We extend Lawrence's representations of the braid groups to relative homology modules and we show that they are free modules over a ring of Laurent polynomials. 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 preserves the integral ring of coefficients, the action of ${U_q{mathfrak{sl}(2)}}$, the braid group representation and its grading. This recovers an integral version for Kohno's theorem relating absolute Lawrence representations with the quantum braid representation on highest-weight vectors. This 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.
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