The paper presents a novel approach for computing the singular homology and higher homotopy groups of finite directed graphs (digraphs) more efficiently.
The key insight is that the directed Vietoris-Rips complex of a finite digraph is weakly homotopy equivalent to the original digraph. This allows replacing the digraph with a finite combinatorial structure (the directed Vietoris-Rips complex) to compute its algebraic topological invariants, despite the associated chain groups being infinite dimensional.
The authors first extend classical results on homotopy and singular homology from topological spaces to the broader setting of pseudotopological spaces, which include digraphs as a full subcategory. This lays the groundwork for proving the main result.
The main theorem states that for each finite digraph, there exists a finite abstract simplicial complex (the directed Vietoris-Rips complex) and a weak homotopy equivalence between its geometric realization and the original digraph. This implies that the singular homology groups of the digraph can be efficiently computed from the finite combinatorial structure of the directed Vietoris-Rips complex.
The authors also prove that weak homotopy equivalences induce isomorphisms on singular (co)homology groups, further justifying the utility of their main result. Overall, this work provides a novel approach for studying higher homotopy and homology groups of discrete mathematical structures like graphs and digital images.
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by Niko... lúc arxiv.org 09-30-2024
https://arxiv.org/pdf/2409.01370.pdfYêu cầu sâu hơn