Computing the Homology Functor on Semi-algebraic Maps and Diagrams

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map \(f:X \rightarrow Y\) $f:X\to Y$ between closed and bounded semi-algebraic sets. For every fixed \(\ell \ge 0\) $\ell \ge 0$ we give an algorithm with singly exponential complexity that computes bases of the homology groups \(\text{ H}_i(X), \text{ H}_i(Y)\) ${\text{H}}_i(X),{\text{H}}_i(Y)$ (with rational coefficients) and a matrix with respect to these bases of the induced linear maps \(\text{ H}_i(f):\text{ H}_i(X) \rightarrow \text{ H}_i(Y), 0 \le i \le \ell \) ${\text{H}}_i(f):{\text{H}}_i(X)\to {\text{H}}_i(Y),0\le i\le \ell$ . We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.