Subject: Question about topological groups
From: Johannes Ebert
Date: Mon, 13 Feb 2006 19:41:06 +0100 (CET)
Hello,
I have an innocent question on topological groups. A theorem of Morre
states that if G is a commutative topological group, then it is homotopy
equivalent to a product of Eilenberg-Maclane spaces, namely
G \simeq \prod_{i=0}^{\infty} K(\pi_i G,i).
Question: Is it true that this splitting is functorial? More precisely: If
f:G \to H is a group homomorphism (H also abelian), then f is homotopic to
\prod_{i=0}^{\ifty} \pi_{i}(f) under the decomposition above?
For my purpose, it would suffice to assume that f is an automorphism of G.
Best regards,
Johannes Ebert