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A closed 3-manifold M is said to be hyperelliptic if it has an involution τ such that the quotient space of M by the action of τ is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnár [12] are hyperelliptic. Then we determine the isometry groups of such manifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with rst homology group Z 35 (constructed by I. Prok in [16], on the basis of a principal algorithm due to Emil Molnár [13], and by Richardson and Rubinstein in [18]) are also hyperelliptic. Main resultsA complete connected Riemannian n-manifold of constant sectional curvature is briey called n-space form. It is well-known [26] that each hyperbolic n-space form can be represented as an orbit space H n /Γ, where H n is the hyperbolic n-space and Γ is a discrete torsion free subgroup of the isometry group of H n . Of course, Γ is isomorphic to the fundamental group of the quotient manifold H n /Γ. A closed connected n-manifold M n is said to be hyperelliptic if it has an involution τ such that the quotient space M n / τ is topologically homeomorphic to the n-sphere S n . The map τ is called a

In this paper we prove that the parabolic Kazhdan-Lusztig polynomials and the parabolic R-polynomials of the universal Coxeter group can be computed in a combinatorial way, by using special matchings.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.