Wikipedia on Eilenberg-Mac Lane spaces

Wikipedia on Eilenberg-Mac Lane spaces

[Fred E.J. Linton]

Something in Wikipedia on E.-M. spaces I think they've got not quite right.
The article in question: 

http://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space .

The problem: after stating (more or less correctly) that "An important 
property of K(G,n) is that, for any abelian group G, and any CW-complex X, 
the set

     [X, K(G,n)]

of homotopy classes of maps from X to K(G,n) is in natural bijection with  
the n-th singular cohomology group

     H^n(X; G)"

the article goes on to say (incorrectly) that "Since H^n(K(G,n); G) = 
Hom(G,G), there is a distinguished element u {\in} H^n(K(G,n);G) 
corresponding to the identity."

Seems to me all that's justified here would be that 'the set

     [K(G,n), K(G,n)]

of homotopy classes of maps from K(G,n) to itself is in natural bijection  
with H^n(K(G,n); G)', whence "there is a distinguished element u {\in} 
H^n(K(G,n);G) corresponding to the identity."

What exact role Hom(G,G) may have to play here might be of interest in its 
own right, but there's no groundwork for that laid anywhere in this Wiki 
article, and it's not germane to the Yoneda lemma instance being invoked.

Or am I missing something?

In any event, I haven't the optimism or the enthusiasm to care to try 
to revise this Wiki's text -- but I welcome any reader who has to do so.

Cheers, -- Fred



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