arXiv subjects

Sungbok Hong

Publications and source records attributed to Sungbok Hong.

4 recordsLinked to original sources

arXiv2011

Diffeomorphisms of Elliptic 3-Manifolds

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. Our main results are 1. The Smale Conjecture holds for all elliptic 3-manifolds containing geometrically incompressible Klein bottles. These include all quaternionic and prism manifolds. 2. The Smale Conjecture holds for all lens spaces L(m,q) with m at least 3. These results complete the Smale Conjecture for all cases except the 3-dimensional real projective space and those admitting a Seifert fibering over the 2-sphere with three exceptional fibers of types (2,3,3), (2,3,4), or (2,3,5). The technical work needed for these results includes the result that if V is a Haken Seifert-fibered 3-manifold, then apart from a small list of known exceptions, the inclusion from the space of fiber-preserving diffeomorphisms of V to the full diffeomorphism group is a homotopy equivalence. This has as a consequence: 3. The space of Seifert fiberings of V has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background material on diffeomorphism groups is included.

math.GT↗

arXiv2004

The Smale Conjecture for lens spaces

The original Smale Conjecture asserted that the inclusion of the group O(4) of isometries of the round 3-sphere S into the full diffeomorphism group Diff(S) is a homotopy equivalence. The (Generalized) Smale Conjecture asserts that the inclusion of Isom(M) into Diff(M) is a homotopy equivalence whenever M is an elliptic 3-manifold, that is, a closed Riemannian 3-manifold of constant positive curvature. We prove the Smale Conjecture for all lens spaces L(m,q), where m is at least 3.

math.GT↗

arXiv1998

Concentration points for Fuchsian groups

A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point is a conical limit point, but even for finitely generated groups not every conical limit point is a concentration point. A slightly weaker concentration condition is given which is satisfied if and only if p is a conical limit point, but not all conical limit points satisfy it. Examples are given that clarify the relations between various concentration conditions.

math.GT↗

arXiv1997

Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was proven by J. Harer for 2-manifolds and by the second author for Haken 3-manifolds. In this paper we prove that H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are proven for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in the boundary of M, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.

math.GT↗