Weeks manifold

In mathematics, the Weeks manifold, sometimes called the Fomenko-Matveev-Weeks manifold, is a closed hyperbolic 3-manifold obtained by (5,2) and (5,1) Dehn surgeries on the Whitehead link. It has volume approximately equal to .9427... and has the smallest known volume of any hyperbolic 3-manifold to date (as of February 2006). The manifold was independently discovered by Jeff Weeks, and also Sergei Matveev and Anatoly Fomenko, working together.

Since the Weeks manifold is an arithmetic hyperbolic 3-manifold, its volume can be computed using its arithmetic data and a formula due to A. Borel:$frac\{3\; cdot23^\{3/2\}zeta\_k(2)\}\{4pi^4\}$,where k is the number field generated by $heta$ satisfying $heta^3-\; heta+1=0$ and $zeta\_k$ is the Dedekind zeta function of k (Chinburg, et al 2001).

Note that the cusped hyperbolic 3-manifold obtained by (5,1) Dehn surgery on the Whitehead link is the so-called sibling manifold, or sister, of the figure eight knot complement. The figure eight knot's complement and its sibling are known to have the smallest volume of any orientable, cusped hyperbolic 3-manifold. Thus if the Weeks manifold were the smallest closed hyperbolic 3-manifold, this shows that the smallest such manifold can be obtained by hyperbolic Dehn surgery on the smallest orientable cusped hyperbolic 3-manifold.

As of February 2006, the best lower bound for the minimal volume closed orientable hyperbolic 3-manifold is .67, due to Ian Agol and Nathan Dunfield. Previously the best lower bound was .33, due to Andrew Przeworski. In June 2007, Gabai, Meyerhoff, and Milley put a preprint on the arXiv claiming to prove the Weeks manifold was the smallest orientable hyperbolic 3-manifold.

References

*Ian Agol, Nathan M. Dunfield, Peter A. Storm, William Thurston."Lower bounds on volumes of hyperbolic Haken 3-manifolds." [http://front.math.ucdavis.edu/math.DG/0506338 arXiv:math.DG/0506338]
*Chinburg, Ted; Friedman, Eduardo; Jones, Kerry N.; Reid, Alan W. "The arithmetic hyperbolic 3-manifold of smallest volume." Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 1-40.
* Sergei V. Matveev and Anatoly T. Fomenko, "Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds." (Russian) Uspekhi Mat. Nauk 43 (1988), no. 1(259), 5--22, 247; translation in Russian Math. Surveys 43 (1988), no. 1, 3--24 MathSciNet| id = 0937017
* Gabai, Meyerhoff, and Milley; "Minimal volume cusped hyperbolic three-manifolds," [http://front.math.ucdavis.edu/0705.4325 arXiv preprint]
*Jeffrey Weeks, "Hyperbolic structures on 3-manifolds", Princeton Univ. Ph.D. thesis, 1985