Computational Topology in a Collapsing Universe: Laplacians, Homology, Cohomology

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 R d+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 dim H1(K) = 0 and K ⊂ X for X a collapsible simplicial complex embedded in R 3 with a known collapsing sequence. Our algorithm runs in Õ(n log 2 (nκ/ε)) 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 ε 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(n d ) time, where n d 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(β d-1 n d ) time; β d-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 R 3 in O(n d log n d + β d-1 n) time using a homology basis computed by the algorithm of Dey [SODA 2019].

For all of the problems above, if K ⊂ R 3 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.