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Developing an algorithm for computing the Betti numbers of semi-algebraicsets with singly exponential complexity has been a holy grail in algorithmicsemi-algebraic geometry and only partial results are known. In this paper weconsider the more general problem of computing the image under the homologyfunctor of a semi-algebraic map $f:X \rightarrow Y$ between closed and boundedsemi-algebraic sets. For every fixed $\ell \geq 0$ we give an algorithm withsingly exponential complexity that computes bases of the homology groups$\mathrm{H}_i(X), \mathrm{H}_i(Y)$ (with rational coefficients) and a matrixwith respect to these bases of the induced linear maps$\mathrm{H}_i(f):\mathrm{H}_i(X) \rightarrow \mathrm{H}_i(Y), 0 \leq i \leq\ell$. We generalize this algorithm to more general (zigzag) diagrams of mapsbetween closed and bounded semi-algebraic sets and give a singly exponentialalgorithm for computing the homology functors on such diagrams. This allows usto give an algorithm with singly exponential complexity for computing barcodesof semi-algebraic zigzag persistent homology in small dimensions.

Computing the homology functor on semi-algebraic maps and diagrams