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Apr 2 at 12:51 comment added S. carmeli Isn't the Cartan homomorphism the map induced on the version of K-theory that also splits short exact sequences? from the way the question is phrased I would guess the question is about the group-completion K-theory, which only formally add direct-sum-inverses but do not splits exact sequences. If this is indeed the intention, this homomorphism is almost never surjective. This comment applies (and even more relevant) to all other answers as well.
Aug 19, 2012 at 19:08 vote accept W. Politarczyk
Aug 18, 2012 at 11:28 history answered Jim Humphreys CC BY-SA 3.0