What is the connection between the mapping class group of a torus and Gl(2,Z)?

In summary, the mapping class group of the torus T^2, which is the group of orientation-preserving self-diffeomorphisms up to isotopy, is isomorphic to Gl(2,Z), the group of orientation-preserving automorphisms. This connection can be seen through the fact that Pi_1(T^2) is isomorphic to Z(+)Z. The connection between automorphisms and matrices is further explored in the book "Knots and Links" by Rolfsen, where a series of exercises can be found.
  • #1
Bacle
662
1
Hi, everyone:

I am trying to understand why the mapping class group of the
torus T^2 (i.e., the group of orientation-preserving self-
diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just
realized this is the name of the group of orientation-preserving
automorphisms). Anyone know what the connection is, between these
two.(or, even better, a proof, or ref. for a proof.)? .
All I can think off is that there may be some connection
with the fact that Pi_1(T^2)=Z(+)Z , but that is all I have.

Thanks for any Ideas.
 
Physics news on Phys.org
  • #2
O.K. I think I made a small break, by finding out (and I came close
to proving) that Aut(Z(+)Z) --which is Pi_1(T^2) --is generated by
three matrices. Now I am trying to find a correspondence between
automorphisms and matrices.
 
  • #3
Check out "Knots and Links" by Rolfsen...I think he has a series of exercises that lead you through it.
 

What is the Mapping Class Group of Torus?

The Mapping Class Group of Torus is a mathematical concept that describes the different ways in which a torus (a doughnut-shaped surface) can be mapped onto itself without tearing or stretching.

How is the Mapping Class Group of Torus studied?

The Mapping Class Group of Torus is studied using algebraic topology, which uses algebraic techniques to study the properties of geometric spaces such as the torus.

What are the applications of the Mapping Class Group of Torus?

The Mapping Class Group of Torus has applications in various fields such as differential geometry, physics, and computer science. It is also used in understanding the geometry of surfaces and in the study of knot theory.

What are the elements of the Mapping Class Group of Torus?

The elements of the Mapping Class Group of Torus are homeomorphisms, which are transformations that preserve the topological properties of the torus.

How does the Mapping Class Group of Torus relate to other mathematical concepts?

The Mapping Class Group of Torus is closely related to other mathematical concepts such as the fundamental group, the homotopy group, and the fundamental groupoid. It is also connected to the theory of Teichmüller spaces, which studies the moduli space of Riemann surfaces.

Similar threads

  • Differential Geometry
Replies
2
Views
3K
  • Differential Geometry
Replies
4
Views
4K
Replies
1
Views
2K
Replies
27
Views
3K
  • Linear and Abstract Algebra
Replies
3
Views
687
  • Differential Geometry
Replies
4
Views
1K
  • Differential Geometry
Replies
5
Views
3K
  • Linear and Abstract Algebra
Replies
17
Views
4K
  • Topology and Analysis
Replies
9
Views
2K
  • Differential Geometry
Replies
2
Views
3K
Back
Top