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Cubical approximation and computation of homology

William Kalies, Konstantin Mischaikow, Greg Watson (1999)

Banach Center Publications

The purpose of this article is to introduce a method for computing the homology groups of cellular complexes composed of cubes. We will pay attention to issues of storage and efficiency in performing computations on large complexes which will be required in applications to the computation of the Conley index. The algorithm used in the homology computations is based on a local reduction procedure, and we give a subquadratic estimate of its computational complexity. This estimate is rigorous in two...

Curiosités Lagrangiennes en dimension 4

Denis Sauvaget (2004)

Annales de l’institut Fourier

Dans ce texte, on définit, pour les immersions lagrangiennes de variétés fermées dans ℂ n , une notion d’aire symplectique enlacée. Puis on construit, dans le cas n = 2 , un certain nombre de surfaces lagrangiennes enlaçant une aire infinie. Dans le cas des surfaces exactes, elles ont le minimum de points doubles possible permis par la théorie (sauf la sphère), c’est-à-dire moins que prévu par quelques conjectures.

Cycles on algebraic models of smooth manifolds

Wojciech Kucharz (2009)

Journal of the European Mathematical Society

Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M . We study modulo 2 homology classes represented by algebraic subsets of X , as X runs through the class of all algebraic models of M . Our main result concerns the case where M is a spin manifold.

Cyclic branched coverings and homology 3-spheres with large group actions

Bruno P. Zimmermann (2004)

Fundamenta Mathematicae

We show that, if the covering involution of a 3-manifold M occurring as the 2-fold branched covering of a knot in the 3-sphere is contained in a finite nonabelian simple group G of diffeomorphisms of M, then M is a homology 3-sphere and G isomorphic to the alternating or dodecahedral group 𝔸₅ ≅ PSL(2,5). An example of such a 3-manifold is the spherical Poincaré sphere. We construct hyperbolic analogues of the Poincaré sphere. We also give examples of hyperbolic ℤ₂-homology 3-spheres with PSL(2,q)-actions,...

Cyclic branched coverings of 2-bridge knots.

Alberto Cavicchioli, Beatrice Ruini, Fulvia Spaggiari (1999)

Revista Matemática Complutense

In this paper we study the connections between cyclic presentations of groups and the fundamental group of cyclic branched coverings of 2-bridge knots. Then we show that the topology of these manifolds (and knots) arises, in a natural way, from the algebraic properties of such presentations.

Cyclic branched coverings of knots and homology spheres.

Francisco González-Acuña, Hamish Short (1991)

Revista Matemática de la Universidad Complutense de Madrid

We study cyclic coverings of S3 branched over a knot, and study conditions under which the covering is a homology sphere. We show that the sequence of orders of the first homology groups for a given knot is either periodic of tends to infinity with the order of the covering, a result recently obtained independently by Riley. From our computations it follows that, if surgery on a knot k with less than 10 crossings produces a manifold with cyclic fundamental group, then k is a torus knot.

Cyclic homology and equivariant theories

Jean-Luc Brylinski (1987)

Annales de l'institut Fourier

In this article, we present two possible extensions of the classical theory of equivariant cohomology. The first, due to P. Baum, R. MacPherson and the author, is called the “delocalized theory". We attempt to present it in very concrete form for a circle action on a smooth manifold. The second is the cyclic homology of the crossed- product algebra of the algebra of smooth functions on a manifold, by the convolution algebra of smooth functions on a Lie group, when such Lie group act on the manifold....

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