Computing the homology functor on semi-algebraic maps and diagrams

Abstract

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 semi-algebraic map f : X Ñ Y between closed and bounded semi-algebraic sets. For every fixed ě 0 we give an algorithm with singly exponential complexity that computes bases of the homology groups H i pXq, H i pY q (with rational coefficients) and a matrix with respect to these bases of the induced linear maps H i pf q : H i pXq Ñ H i pY q, 0 ď i ď . We generalize this algorithm to more general (zigzag) diagrams of 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.