Central Z extensions
Here's one of my favorite basic algebra/topology things. You can read about this in Brown's Cohomology of Groups book. A short exact sequence of groups $$0 \to \mathbb{Z} \to H \to G \to 1$$ is called a central $\mathbb{Z}$-extension of $G$ if the $\mathbb{Z}$ subgroup of $H$ sits in the center of $H$. For example, $$0 \to \mathbb{Z} \to G \oplus \mathbb{Z} \to G \to 1$$ is the trivial central $\mathbb{Z}$-extension of $G$ and $$0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2 \to 0$$ is a non-trivial extension, where the first map is multiplication by 2. Are there other extensions of $\mathbb{Z}/2$? How do you classify central $\mathbb{Z}$-extensions? Given an extension and a homomorphism $f: G' \to G$, can we lift $f$ to $H$? These seem hard to answer, but actually they are easy!!! And we can topologize this problem. The one thing we need is group cohomology. Lightning crash course for this post: if $G$ is a group, then $H^*(G)$ is just defined to be $H