Author(s): | Krčál, Marek; Pilarczyk, Paweł L |

Title: | Computation of cubical Steenrod squares |

Title Series: | LNCS |

Affiliation | IST Austria |

Abstract: | Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. 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 → Sd to a sphere 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 f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem. |

Keywords: | Chain contraction; Cohomology operation; Cubical complex; Cup product |

Conference Title: | CTIC: Computational Topology in Image Context |

Volume: | 9667 |

Conference Dates: | June 15 - 17, 2016 |

Conference Location: | Marseille, France |

ISBN: | 978-331939440-4 |

Publisher: | Springer |

Date Published: | 2016-01-01 |

Start Page: | 140 |

End Page: | 151 |

Sponsor: | 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.). |

DOI: | 10.1007/978-3-319-39441-1_13 |

Open access: | no |