Subject: Question about fundamental groups of topological groups
From: Masoud
Date: Mon, 26 Feb 2007 23:07:00 -0600
Let G be a connected topological group. Is the following complex exact?
\pi_1(G\times G) ----> \pi_1([G,G])---->\pi_1(G)
Here all the fundamental groups have as their base points the identity
elements of the respective groups. The first map is induced by the
commutator map
(g,h) \mapsto ghg^{-1}h^{-1},
the second map is induced by the natural inclusion.
Warning: The commutator map is not a group morphism.
Thank you,
Masoud
Graduate Student
University of Chicago