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We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the 3 3-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how the algorithm can be used to infer information about the fundamental group π _1 (K) π 1 (K) of a metric space K K using only a finite point cloud X X sampled from the space. In the special case where K K is a d d-dimensional compact manifold K ⊂\mathbb R^ d K⊂ R d, we consider the closure of the complement of K K in the d d-sphere M_K=\mathbb S^ d\! ∖\! K MK= S d\K¯. For a base-point x x in the boundary ∂ M_K∂ MK of the manifold M_K MK one can attempt to determine, from the point cloud X X, the induced homomorphism of fundamental groups ϕ: π _1 (∂ …