Abstract:
We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling theorem it is then natural to face the problem of classifying all the exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the symmetries of M5 and of some hyperbolic manifolds resulting from fillings of M5, the set of exceptional fillings on M5 is described by a very small amount of information.

Abstract:
For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.

Abstract:
We study the algebraic and geometric structures for closed orientable 3-manifolds obtained by Dehn surgery along the family of hyperbolic links with certain surgery coefficients and moreover, the geometric presentations of the fundamental group of these manifolds. We prove that our surgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link by applying the Montesinos theorem in Montesinos-Amilibia (1975). In particular, our result includes the topological classification of the closed 3-manifolds obtained by Dehn surgery on the Whitehead link, according to Mednykh and Vesnin (1998), and the hyperbolic link

Abstract:
In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair (R,K), we consider the knot K^n gotten by twisting K n times along R and give a lower bound on the bridge number of K^n with respect to Heegaard splittings of M -- as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of K^n tends to infinity with n. In application, we look at a family of knots K^n found by Teragaito that live in a small Seifert fiber space M and where each K^n admits a Dehn surgery giving the 3-sphere. We show that the bridge number of K^n with respect to any genus 2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving the 3-sphere.

Abstract:
When can one 3-manifold be transformed to another by a finite sequence of Dehn surgeries which are restricted to preserve the first homology of the manifolds ? What is the resulting equivalence relation on 3-manifolds ? What if the surgery circle is further restricted to lie more deeply in the lower central series of the fundamental group ? We answer these questions. It is shown that many of these questions have answers in terms of classical toplogical invariants. Relations with Heegard splittings and the Torelli group are discussed. This is also related to whether or not one 3-manifold may be obtained from another by Dehn surgery on a link of restricted type. These equivalence relations form the philosophical basis of the authors joint work with Paul Melvin on a theory of finite type invariants for arbitrary 3-manifolds.

Abstract:
Consider a three dimensional cusped spherical $\mathrm{CR}$ manifold $M$ and suppose that the holonomy representation of $\pi_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{CR}$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz's spherical $\mathrm{CR}$ Dehn surgery theorem, but has weaker hypotheses and does not give the unifomizability of the structure. We apply our theorem in the case of the Deraux-Falbel structure on the Figure Eight knot complement and obtain spherical $\mathrm{CR}$ structures on all Dehn surgeries of slope $-3 + r$ for $r \in \mathbb{Q}^{+}$ small enough.

Abstract:
This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings $M(\alpha)$ and $M(\beta)$ on a given 3-manifold $M$, each containing a surface that is either essential or a Heegaard surface. We show how a combinatorial analysis of the graphs of intersection of the two corresponding punctured surfaces in $M$ enables one to find faces of these graphs which give useful topological information about $M(\alpha)$ and $M(\beta)$, and hence, in certain cases, good upper bounds on the intersection number $\Delta(\alpha, \beta)$ of the two filling slopes.

Abstract:
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic cone-manifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.

Abstract:
The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2\pi$ theorem by showing that if each filling slope has length more than six, then the resulting 3-manifold has all the above properties. We also give a combinatorial version of the $2\pi$ theorem which relates to angled ideal triangulations. We apply these techniques by studying surgery along alternating links.

Abstract:
We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients.