#### Volume 6, issue 5 (2006)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Dehn surgery, homology and hyperbolic volume

### Ian Agol, Marc Culler and Peter B Shalen

Algebraic & Geometric Topology 6 (2006) 2297–2312
 arXiv: math.GT/0508208
##### Abstract

If a closed, orientable hyperbolic $3$–manifold $M$ has volume at most 1.22 then ${H}_{1}\left(M;{ℤ}_{p}\right)$ has dimension at most $2$ for every prime $p\ne 2,7$, and ${H}_{1}\left(M;{ℤ}_{2}\right)$ and ${H}_{1}\left(M;{ℤ}_{7}\right)$ have dimension at most $3$. The proof combines several deep results about hyperbolic $3$–manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic $C\subset M$ with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from $M$ by Dehn surgeries on $C$.

##### Keywords
hyperbolic manifold, volume, homology, drilling, Dehn surgery
Primary: 57M50
Secondary: 57M27