Computation of Cubical Steenrod Squares
Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in Open image in new window . 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. González-Díaz, 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 \(A\rightarrow S^d\) to a sphere \(S^d\) be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function Open image in new window and a value \(r>0\) whether every Open image in new window with \(\Vert g-f\Vert _\infty \le r\) has a root, reduces to the extension problem.
KeywordsCohomology operation Cubical complex Cup product Chain contraction
The research conducted by both authors has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreements no. 291734 (for M. K.) and no. 622033 (for P. P.).
- 2.Computational Homology Project software. http://chomp.rutgers.edu/software/
- 3.Computer Assisted Proofs in Dynamics group. http://capd.ii.uj.edu.pl/
- 10.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001). http://www.math.cornell.edu/ hatcher/AT/ATpage.html MATHGoogle Scholar
- 15.Franek, P., Krčál, M., Wagner, H.: Robustness of zero sets: Implementation, submittedGoogle Scholar
- 16.Pilarczyk, P.: The ChainCon software. Chain contractions,homology and cohomology software and examples. http://www.pawelpilarczyk.com/chaincon/
- 18.Prasolov, V.V.: Elements of Homology Theory. Graduate Studies in Mathematics, American Mathematical Society (2007)Google Scholar
- 20.Sergeraert, F.: Effective homology, a survey (1992). http://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Survey.pdf
- 25.Vokřínek, L.: Decidability of the extension problem for maps into odd-dimensional spheres. [math.AT] (2014). arXiv:1401.3758