|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
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)
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? 220.127.116.11 (talk) 03:23, 8 July 2013 (UTC)
- "Building blocks" --> general toplogical spaces can be built via Postnikov system aka the Postnikov tower. "Intuitive" --> not so much. Notice that all but one of the examples is K(G,1), the only other example is CP^\infty which is KG(Z,2) which follows from the Hopf fibration for the finite-dimensional CP's. You can build more as loop spaces. Not sure if loop spaces count as either intuitive or elementary. I mean, they sound simple, at first, until you dive in deeper, and then suddenly you're doing M-theory, wtf. I'm saying this only because maybe someone else can provide a more "elementary" and "intuitive" explanation. 18.104.22.168 (talk) 05:12, 6 May 2016 (UTC)
- It doesn't redirect to itself -- note that your link contains a long dash, while the actual page name contains a short dash. - Saibod (talk) 12:17, 28 May 2014 (UTC)
It's a minor detail, but... "Let G be a group and n a positive integer. A connected topological space X is called..." Should that be simply connected? Otherwise it might not have a single homotopy group. 22.214.171.124 (talk) 01:31, 18 April 2016 (UTC)
- Yes, but you don't actually have to say that, as its already built into the definition: K(G,n) for n>1 implies that only is non-trivial, i.e. it implies that all other pi's such as are trivial, i.e. its simply connected. Mathematicians are cleverly parsimonious. I'm not sure exactly how to add that to the article, as its a forehead-slapper once you see it. 126.96.36.199 (talk) 04:20, 6 May 2016 (UTC)