Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set S Rk by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex Δ, whose geometric realization, |Δ|, is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any ℓ ≥ 0, there exists an algorithm which takes as input a description of a semialgebraic subset Sc Rk given by a quantifier-free first-order formula ϕ in the language of the reals and produces as output a simplicial complex Δ, whose geometric realization, |Δ| is ℓ-equivalent to S. The complexity of our algorithm is bounded by (sd)k0(ℓ), where s is the number of polynomials appearing in the formula ϕ, and d a bound on their degrees. For fixed ℓ, this bound is singly exponential in k. In particular, since ℓ-equivalence implies that the homotopy groups up to dimension ℓ of |Δ| are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first ℓ homotopy groups of S to the combinatorial problem of computing the first ℓ homotopy groups of a finite simplicial complex of size bounded by (sd)k0(ℓ). [ABSTRACT FROM AUTHOR]