|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
More information, please. In particular, in what sense are these spaces "building blocks for homotopy theory"? And, can this topic be introduced in more elementary terms to give some intuition? 22.214.171.124 (talk) 03:23, 8 July 2013 (UTC)
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)
- Duly noted – I’ve added a note to that effect.
- —Nils von Barth (nbarth) (talk) 17:26, 24 July 2009 (UTC)
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)