Computation of Cubical Steenrod Squares

Abstract

Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in [InlineEquation not available: see fulltext.]. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. Gonzalez-Diaz, Real, Homology, Homotopy Appl, 2003, 83---93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253---275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290---320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map $$A\rightarrow S^d$$Ai¾?Sd to a sphere $$S^d$$Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function [InlineEquation not available: see fulltext.] and a value $$r>0$$r>0 whether every [InlineEquation not available: see fulltext.] with $$\Vert g-f\Vert _\infty \le r$$i¾?g-fi¾?∞≤r has a root, reduces to the extension problem.