Assignment:
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).
Provide complete and step by step solution for the question and show calculations and use formulas.