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2016-01-07 17:30:57 Post No. 7767928
Post No. 7767928
Let's have an algebraic topology thread.
Yesterday I learned about Eilenberg-MacLane spaces. That is for a group G and a natural number n you can construct a topological space K(G,n) such that G is the n-th homotopy group and all other homotopy groups are trivial.
Can I get a similar result if I want more homotopy groups to be non-trivial and to equal other previously specified groups? For example if I have a set of groups (Gi) with i in some finite or countably infinite index set, can I find a space where the i-th homotopy group is Gi for all i? Or is there some obstruction making such a space impossible?