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Register a new userFundamental Group Algorithm for low dimensional tessellated CW complexes
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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.
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If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Walls D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincar{e} $3$-complex $X$ with $pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $mathbb{Z} G$ modules where $G$ has periodic cohomology.
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chens iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a CDGA suitable for integration. We show that this integration is compatible with Bott-Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.
In this article we lay out the details of Fukayas $A_infty$-structure of the Morse complexe of a manifold possibly with boundary. We show that this $A_infty$-structure is homotopically independent of the made choices. We emphasize the transversality arguments that make some fiber products smooth.
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
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
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