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Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraicclosure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraicbasis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, withcoefficients in a field $\mathbb{F}$, of any given semi-algebraic set $S\subset \mathrm{R}^k$ defined by a closed formula. The complexity of thealgorithm is bounded singly exponentially. It is not known how to compute sucha basis for the higher homology groups with singly exponential complexity.  As an intermediate step in our algorithm we construct a semi-algebraic subset$\Gamma$ of the given semi-algebraic set $S$, such that $\mathrm{H}_q(S,\Gamma)= 0$ for $q=0,1$. We relate this construction to a basic theorem in complexalgebraic geometry stating that for any affine variety $X$ of dimension $n$,there exists Zariski closed subsets \[ Z^{(n-1)} \supset \cdots \supset Z^{(1)}\supset Z^{(0)} \] with $\dim_{\mathrm{C}} Z^{(i)} \leq i$, and$\mathrm{H}_q(X,Z^{(i)}) = 0$ for $0 \leq q \leq i$. We conjecture aquantitative version of this result in the semi-algebraic category, with $X$and $Z^{(i)}$ replaced by closed semi-algebraic sets. We make initial progresson this conjecture by proving the existence of $Z^{(0)}$ and $Z^{(1)}$ withcomplexity bounded singly exponentially (previously, such an algorithm wasknown only for constructing $Z_0$).

Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set