# Three-Dimensional Topological Group

Please help with the following problem.

Let M = SL(2) be the set of 2 × 2 matrices with unit determinant. Show that,

when regarded as a subset of R4 under

( a b )

( c d ) <--> (a, b, c, d) Exists R4 and equipped with subspace topology, SL(2) becomes a 3-dimensional topological group. That is, show that (i) SL(2) is a group under matrix multiplication, (ii) SL(2) is a 3-

manifold (find coordinate maps!), and (iii) the multiplication operation (A,B)--> AB−1 is continuous as a mapping from SL(2) × SL(2) ---> SL(2).

https://brainmass.com/math/geometry-and-topology/three-dimensional-topological-group-235796

#### Solution Preview

Problem: Let M = SL(2) be the set of 2 × 2 matrices with unit determinant. Show that,

when regarded as a subset of R4 under

( a b )

( c d ) <--> (a, b, c, d) Exists R4 and equipped with subspace topology, SL(2) becomes a 3-dimensional topological group. That is, show that (i) SL(2) is a group under matrix multiplication, (ii) SL(2) is a 3-

manifold (find coordinate maps!), and (iii) the multiplication ...

#### Solution Summary

This solution helps with a problem involving a 3D topological group.