Eilenberg–Mac Lane Spaces

Prerequisites. Direct starred prerequisites: Chapter 71. No later core chapter depends on this route.

Opening obstruction

Ordinary groups occur as fundamental groups, but cohomology requires spaces with one prescribed nontrivial homotopy group in any degree.

Development contract

Construct $K(G,1)$ as the connected one-object delooping of a group $G$, and construct $K(A,n)$ for abelian $A$ and $n\ge 2$ by iterated delooping. Prove $\mathrm{\Omega }K(A,n+1)\simeq K(A,n)$ with the connectedness and truncation bounds that identify its sole nontrivial homotopy group.