Computational Topology in a Collapsing Universe: Laplacians, Homology, Cohomology
Mitchell Black, William Maxwell, Amir Nayyeri, Eli Winkelman · Society for Industrial and Applied Mathematics eBooks · 2022
We consider a variety of topology problems on a d-dimensional simplicial complex K given that K ∪ X for X a collapsible simplicial complex embedded in ℝd+1 with known collapsing sequence. Our first result is a solver for the linear system L1x = b, where L1 is the 1-Laplacian of a simplicial complex K with dimH1(K) = 0 and K ∪ X for X a collapsible simplicial complex embedded in ℝ3 with a known collapsing sequence. Our algorithm runs in O(n log2 (nκ/∊)) time, where n is the total number of vertices, edges, and triangles in X, κ is the largest condition number of the two parts of the Laplacian, and ∊ quantifies the approximation quality. This result is a generalization of Cohen et al. [SODA 2014]. The new technical piece of our Laplacian solver, in addition to the machinery described by Cohen et al., is an algorithm to compute a bounding chain of a 1-cycle within k. In addition, we describe faster algorithms for testing null-homology of (d–1)-cycles and null-cohomology of d-cocycles. Our algorithm runs in O(nd) time, where nd is the number of d-simplices in X. Finally, we describe an algorithm to compute a (d–1)-cohomology basis from a given (d–1)-homology basis for a d-simplicial complex K in O(βd–1nd) time; βd–1 is the rank of the (d–1)st homology group of k. In particular, we can obtain a cohomology basis for subcomplexes of a collapsible complex X embedded in ℝ3 in O(nd log nd + βd–1 n) time using a homology basis computed by the algorithm of Dey [SODA 2019]. For all of the problems above, if K ∪ ℝ3 and the collapsible supercomplex X is not provided, we can expand K into a convex ball of possibly quadratic complexity, which is known to be collapsible, resulting in nearly quadratic time algorithms.