The Sphere in Three Dimensions and Higher:
Generalizations and Special Cases

Matt Chisholm
matt (at) theory (dot) org

Abstract:

I examine the general mathematical object that is the sphere. First, I examine several constructions general to all spheres. Then, I look at the Clifford Tori, the Hopf Fibration, group structure and surgery in $\Bbb{S}^{3}$ , with respect to the unique structure of $\Bbb{S}^{3}$ . Finally, I examine a unique dissection of $\Bbb{S}^{3}$ which takes advantage of a certain triangulation and the Clifford Tori. This dissection can be used to explain surgery on the unknot in an interesting way.

$\includegraphics{gfx/s3_map.ps}$

Overview of the Sphere
- Definitions
- Construction of $\Bbb{S}^{n}$ by two N-disks
Visualizing the N-Sphere
- The two balls model
- Stereographic Projection
Inversion
- Inverting $\Bbb{S}^{n-1}$ inside $\Bbb{S}^{n}$
- Inverting $\mathrm{\mathbf{T}}^{n-1}$ inside $\Bbb{S}^{n}$
  - Inverting $\mathrm{\mathbf{T}}^{n-1}$ inside $\Bbb{S}^{n}$
Clifford Tori and Hopf Fibration
Triangulation of $\Bbb{S}^{n}$
Surgery on Knots
- Turning $\Bbb{S}^{3}$ into $\ensuremath{\Bbb{S}^{2}} \times \ensuremath{\Bbb{S}^{1}}$ by removing and replacing $\mathrm{\mathbf{T}}^{2}$
  - Separate $\Bbb{S}^{3}$ into two filled $\mathrm{\mathbf{T}}^{2}$
  - Re-glue the tori
A visualization of The Clifford Tori, Triangulation, and Surgery
Conclusion
- Acknowledgements
References
About this document ...

Adric 2001-03-23

The Sphere in Three Dimensions and Higher: Generalizations and Special Cases

Abstract:

The Sphere in Three Dimensions and Higher:
Generalizations and Special Cases