Talk:Eilenberg-MacLane space
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Page name - I know MacLane adopted Saunders Mac Lane; but Eilenberg-MacLane is standard in the literature.
Charles Matthews 14:01, 17 Oct 2004 (UTC)
[edit] Representability
It is certainly not true that for abelian π and any topological space X, the set [X, K(π,n)] of homotopy classes of based maps from X to K(π,n) is in natural bijection with n-th coholmology Hn(X; π) of the space X. A good counterexample is the pseudocircle. The usual correct statement of this result is with X restricted to be a CW complex. However another common formulation is to allow X to be arbitrary, but then take [X, K(π,n)] to be the Hom set in the weak homotopy category. This basically amounts to the same thing, since that Hom set is obtained by replacing X by a CW approximation and taking based homotopy classes.
However I have heard of a formulation of this result where one takes based homotopy classes with X not necessarily a CW complex, but something like a compact metric space. The resulting set of homotopy classes is in 1-1 correspondence with something like Cech cohomology instead of singular cohomology. Does anyone know the precise formulation of this result? Fiedorow 17:43, 16 December 2005 (UTC)
- Never mind. I found a reference for this result and incorporated it into the article. Fiedorow 18:53, 16 December 2005 (UTC)
