Les invariants ${\theta }_p$ des 3-variétés périodiques

Chbili, Nafaa

Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 1135-1150.

Soit $r$ un entier $>1$. Une 3-variété $M$ est dite $r$-périodique si et seulement si le groupe cyclique $G=\mathbb{Z}/r\mathbb{Z}$ agit semi-librement sur $M$ avec un cercle comme l’ensemble des points fixes. Dans cet article, nous utilisons les invariants quantiques ${\theta }_p$ pour établir des conditions nécessaires pour qu’une 3-variété soit périodique.

Let $r\ge 2$ be an integer. A 3-manifold $M$ is said to be $r$-periodic if and only if the group $G=\mathbb{Z}/r\mathbb{Z}$ acts smoothly on $M$ with a circle as the set of fixed points. The aim of this paper is to study the invariants ${\theta }_p\left(M\right)$ in the case where $M$ is an $r$-periodic 3-homology sphere. We use the regularity of the Kauffman bracket of periodic links introduced by Murasugi, to find a relationship between the invariant of $M$ and the invariant of the quotient 3-homology sphere $\overline{M}$. As an application it is shown that the Poincaré space is not the regular $r-$fold branched covering of $S^3$, if $r$ is a prime congruent to $\pm 1$ modulo $5$.

Classification : 57M27
Mot clés : 3-variété périodique, entrelacs périodique, sphère d'homologie, invariants quantiques
Keywords: periodic 3-manifold, periodic link, homology sphere, quantum invariants

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