We present a new algorithm for computing the topology of a real algebraic surface $S$, even in singular cases. We use previous algorithms for 2D and 3D algebraic curves and show how properties of the polar variety of $S$ yield a topological complex equivalent to $S$, or even a simplicial complex isotopic to $S$. The proof of correctness of the algorithm is detailed. It is based on tools from stratification theory. We construct an explicit Whitney stratification of $S$, by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of $S$ from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectivity issues conclude the paper.
