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Realisation problem for the space of knots in the 3-sphere
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ S^3 $](/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ Sym(L) = \pi_0 Diff(S^3,L) $](/files/tex/4864a515a83902b1bc11af895975e9eb26387dda.png)
![$ S^3 $](/files/tex/02a9a17122cd1be0450f9ddf93c53e3feb250aad.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
![$ L $](/files/tex/73f33398d7e8aa42e6ec25ee2bb4f2b57ed3391a.png)
Now let be a hyperbolic link. Assume
has the further `Brunnian' property that there exists a component
of
such that
is the unlink. Let
be the subgroup of
consisting of diffeomorphisms of
which preserve
together with its orientation, and which preserve the orientation of
.
There is a representation given by restricting the diffeomorphism to the
. It's known that
is always a cyclic group. And
is a signed symmetric group -- the wreath product of a symmetric group with
.
Problem: What representations can be obtained?
An answer to this problem would give a `closed form' description of the homotopy type of the space of smooth embeddings of in
. This is the space of embeddings in the Whitney Topology, or
-uniform topology for any
.
`Closed form' means that every component of would have the description as an iterated fiber bundle over certain well-known spaces, where the fibers are inductively well-known spaces, and the monodromy would be controlled rather explicitly by this list of representations.
Peripherally related are various other realization problems for -manifolds. For example, Sadayoshi Kojima proved that one can realize any finite group as the group of isometries of a hyperbolic
-manifold.
Bibliography
*[B] Budney, R. Topology of spaces of knots in dimension 3, to appear in Proc. Lond. Math. Soc.
[B2] Budney, R. A family of embedding spaces. Geometry and Topology Monographs 13 (2007).
[K] Kojima, S., Isometry transformations of hyperbolic -manifolds. Topology Appl. 29 (1988), no. 3, 297--307.
* indicates original appearance(s) of problem.