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Fundamental pro-groupoids and covering projections

Luis Hernández-Paricio (1998)

Fundamenta Mathematicae

We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX,...

Generalized n-colorings of links

Daniel Silver, Susan Williams (1998)

Banach Center Publications

The notion of an (n,r)-coloring for a link diagram generalizes the idea of an n-coloring introduced by R. H. Fox. For any positive integer n the various (n,r)-colorings of a diagram for an oriented link l correspond in a natural way to the periodic points of the representation shift Φ / n ( l ) of the link. The number of (n,r)-colorings of a diagram for a satellite knot is determined by the colorings of its pattern and companion knots together with the winding number.

Generalized universal covering spaces and the shape group

Hanspeter Fischer, Andreas Zastrow (2007)

Fundamenta Mathematicae

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ: π₁(X) → π̌₁(X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if φ: π₁(X) → π̌₁(X) is injective. This generalized notion of universal covering p: X̃ → X enjoys most of the usual properties, with the possible exception of evenly...

Generating series and asymptotics of classical spin networks

Francesco Costantino, Julien Marché (2015)

Journal of the European Mathematical Society

We study classical spin networks with group SU 2 . In the first part, using Gaussian integrals, we compute their generating series in the case where the edges are equipped with holonomies; this generalizes Westbury’s formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.

Genus 2 Heegaard decompositions of small Seifert manifolds

Michel Boileau, D. J. Collins, H. Zieschang (1991)

Annales de l'institut Fourier

The genus 2 Heegaard splittings and decompositions of Seifert manifolds over S with 3 exeptional fibres are classified with respect to isotopies and homeomorphisms. In general there are 3 different isotopy classes of Heegaard splittings and 6 different isotopy classes of Heegaard decompositions. Moreover, we determine when a homeomorphism class is not an isotopy class.

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