Exotic group actions on homology 3-spheres
We present two contrasting methods for constructing group actions on homology 3-spheres. These actions are characterized as being exotic - not conjugate to a standard linear action. The first method is to modify a standard linear action on the 3-sphere by performing surgery on an invariant link....
| Main Author: | |
|---|---|
| Language: | English |
| Published: |
2009
|
| Online Access: | http://hdl.handle.net/2429/13454 |
| Summary: | We present two contrasting methods for constructing group actions on homology
3-spheres. These actions are characterized as being exotic - not conjugate
to a standard linear action.
The first method is to modify a standard linear action on the 3-sphere by
performing surgery on an invariant link. A family of exotic actions is produced
for a class of metacyclic groups. This method is certainly applicable to any
group admitting a linear representation.
The second method is a computer aided study using the program Twiggy,
software written by the author. Starting from a knot in the 3-sphere, a representation
is found from the knot group to a finite group G. The branched cover
associated to the representation has a natural G-action. The program looks for
representations, determines if the cover is a homology sphere and tests if the
action constructed is exotic. The primary test of whether or not an action is
exotic is by examining the linking numbers of the fixed point curves. |
|---|