arXiv subjects

Darryl McCullough

Publications and source records attributed to Darryl McCullough.

25 records · Page 2Linked to original sources

arXiv2000

The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π_1(M) into PSL 2 (C). The components of the interior MP(π_1(M)) of AH(π_1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(π_1(M)) and hence a conjectural topological enumeration of the components of AH(π_1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(π_1(M)) has infinitely many components.

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arXiv2000

Isometries of elliptic 3-manifolds

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

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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.

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arXiv1998

Fiber-preserving diffeomorphisms and imbeddings

Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifolds, and fiber-preserving diffeomorphisms and imbeddings. That is, if M fibers over B, with compact fiber, and V is a vertical submanifold of M, then the restriction from the space FDiff(M) of fiber-preserving diffeomorphisms of M to the space of imbeddings of V into M that take fibers to fibers is locally trivial. Also, the map from FDiff(M) to Diff(B) that takes f to the diffeomorphism it induces on B is locally trivial. The proofs adapt Palais' original approach; the main new ingredient is a version of the exponential map, called the aligned exponential, which has better properties with respect to fiber-preserving maps. Versions allowing certain kinds of singular fibers are proven, using equivariant methods. These apply to almost all Seifert-fibered 3-manifolds. As an application, we reprove an unpublished result of F. Raymond and W. Neumann that each component of the space of Seifert fiberings of a Haken 3-manifold is weakly contractible.

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arXiv1997

Finiteness of classifying spaces of relative diffeomorphism groups of 3-manifolds

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel dM) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.

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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.

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arXiv1997

The Generalized Smale Conjecture for 3-manifolds with genus 2 one-sided Heegaard splittings

The Generalized Smale Conjecture asserts that if M is a closed 3-manifold with constant positive curvature, then the inclusion of the group of isometries into the group of diffeomorphisms is a homotopy equivalence. For the 3-sphere, this was the classical Smale Conjecture proved by A. Hatcher. N. Ivanov proved the Generalized Smale Conjecture for the M which contain a 1-sided Klein bottle and such that no Seifert fibering is nonsingular on the complement of any vertical Klein bottle. We prove it in all remaining cases containing a one-sided Klein bottle, except for the lens space L(4,1).

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