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In this paper, we develop and study the theory of weighted fundamental groups of weighted simplicial complexes. When all weights are 1, the weighted fundamental group reduces to the usual fundamental group as a special case. We also study weight
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We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously establish
We present a detailed description of a fundamental group algorithm based on Formans combinatorial version of Morse theory. We use this algorithm in a classification problem of prime knots up to 14 crossings.
Let $X$ be an algebraic surface of degree $5$, which is considered as a branch cover of $mathbb{CP}^2$ with respect to a generic projection. The surface has a natural Galois cover with Galois group $S_5$. In this paper, we deal with the fundamental groups of Galois covers of degree $5$ surfaces that degenerate to nice plane arrangements; each of them is a union of five planes such that no three planes meet in a line.
We construct a space $mathbb{P}$ for which the canonical homomorphism $pi_1(mathbb{P},p) rightarrow check{pi}_1(mathbb{P},p)$ from the fundamental group to the first v{C}ech homotopy group is not injective, although it has all of the following properties: (1) $mathbb{P}setminus{p}$ is a 2-manifold with connected non-compact boundary; (2) $mathbb{P}$ is connected and locally path connected; (3) $mathbb{P}$ is strongly homotopically Hausdorff; (4) $mathbb{P}$ is homotopically path Hausdorff; (5) $mathbb{P}$ is 1-UV$_0$; (6) $mathbb{P}$ admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) $pi_1(mathbb{P},p)$ naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) $pi_1(mathbb{P},p)$ is locally free.
In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $phi$-weighted coboundary operator induced by a weight function $phi$. Our weight function $phi$ is a generalization of Dawsons weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.
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