65 research outputs found

### 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

### Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group G on a
3-dimensional handlebody V. Whenever G acts freely preserving orientation on a
connected 3-manifold X, there is a G-equivariant imbedding of V into X. There
are choices of X closed and Seifert-fibered for which the image of V is a
handlebody of a Heegaard splitting of X. Provided that the genus of V is at
least 2, there are similar choices with X closed and hyperbolic

### Roots of Dehn twists

D. Margalit and S. Schleimer found examples of roots of the Dehn twist about
a nonseparating curve in a closed orientable surface, that is, homeomorphisms
whose nth power is isotopic to the Dehn twist. Our main theorem gives
elementary number-theoretic conditions that describe the values of n for which
an nth root exists, given the genus of the surface. Among its applications, we
show that n must be odd, that the Margalit-Schleimer roots achieve the maximum
value of n among the roots for a given genus, and that for a given odd n, nth
roots exist for all genera greater than (n-2)(n-1)/2. We also describe all nth
roots having n greater than or equal to the genus.Comment: 15 pages, 6 figure

### 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.Comment: 24 pages, 7 figure

### The space of Heegaard Splittings

For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) =
Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard
splitting (M,F). Its path components are the isotopy classes of Heegaard
splittings equivalent to (M,F). We describe H(M,F) in terms of Diff(M) and the
Goeritz group of (M,F). In particular, for hyperbolic M each path component is
a classifying space for the Goeritz group, and when the (Hempel) distance of
(M,F) is greater than 3, each path component of H(M,F) is contractible. For
splittings of genus 0 or 1, we determine the complete homotopy type (modulo the
Smale Conjecture for M in the cases when it is not known).Comment: Minor rewriting as suggested by referee, no change in mathematical
content. To appear in J. Reine Angew. Mat

### 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.Comment: 43 pages, LaTeX2

### Middle tunnels by splitting

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle
tunnel is a tunnel that is not an upper or lower tunnel for some
(1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot
examples were obtained by Goda, Hayashi, and Ishihara. We generalize their
construction and calculate the slope invariants for the resulting middle
tunnels. In particular, we obtain the slope sequence of the original example of
Goda, Hayashi, and Ishihara.Comment: 20 pages, 11 figure

### Iterated splitting and the classification of knot tunnels

For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle
tunnel is a tunnel that is not an upper or lower tunnel for some
(1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot
examples were obtained by Goda, Hayashi, and Ishihara. In a previous paper, we
generalized their construction and calculated the slope invariants for the
resulting examples. We give an iterated version of the construction that
produces many more examples, and calculate their slope invariants. If one
starts with the trivial knot, the iterated constructions produce all the
2-bridge knots, giving a new calculation of the slope invariants of their
tunnels. In the final section we compile a list of the known possibilities for
the set of tunnels of a given tunnel number 1 knot.Comment: The results of the paper are unchanged. The list of known tunnel
phenomena has been enlarged to include new possibilities seen in examples
recently found by John Berge, after reading the previous version of the
paper. The previous list was presented as a conjecture of all possibilities,
but the new list is presented only as list of known phenomena, prompting the
change of titl

### Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free
actions of a finite group G on an orientable 3-dimensional handlebody of genus
g can be enumerated in terms of sets of generators of G. They correspond to the
equivalence classes of generating n-vectors of elements of G, where
n=1+(g-1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For
abelian and dihedral G, this allows a complete determination of the equivalence
and weak equivalence classes of actions for all genera. Additional information
is obtained for solvable groups and for the groups PSL(2,3^p) with p prime. For
all G, there is only one equivalence class of actions on the genus g handlebody
if g is at least 1+r(G)|G|, where r(G) is the maximal length of a chain of
subgroups of G. There is a stabilization process that sends an equivalence
class of actions to an equivalence class of actions on a higher genus, and some
results about its effects are obtained

### The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic
base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R
or the universal cover of SL(2,R). Our main result is that the connected
component of the identity map in the diffeomorphism group Diff(M) is either
contractible or homotopy equivalent to the circle, according as the center of
the fundamental group of M is trivial or infinite cyclic. Apart from the
remaining case of non-Haken infranilmanifolds, this completes the homeomorphism
classifications of Diff(M) and of the space of Seifert fiberings of M for all
compact orientable aspherical 3-manifolds. We also prove that when the base
orbifold of M is hyperbolic with underlying manifold the 2-sphere with three
cone points, the inclusion from the isometry group Isom(M) to Diff(M) is a
homotopy equivalence

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