For an undirected edge-weighted graph $G$ and a set $R$ of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from $G$ disconnects each pair in $R$. We provide an algorithm computing a $(1+\varepsilon)$-approximation of the minimum multicut of a graph $G$ in time $(g+t)^{(O(g+t)^3)}\cdot(1/\varepsilon)^{O(g+t)} \cdot n \log n$, where $g$ is the genus of $G$ and $t$ is the number of terminals. This result is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when $g=0$). In order to achieve this, our article leverages on a novel characterization of a minimum multicut as a family of Steiner trees in the universal cover of a surface on which $G$ is embedded. The algorithm heavily relies on topological techniques, and in particular on the use of homotopical tools and computations in covering spaces, which we blend with classic ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs.