# Are all degree-1 cohomology operations Bocksteins?

I'm interested in cohomology operations (in ordinary cohomology) $$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$ that is, elements of $$H^{i+1}(K(G, i), H)\;.$$ I know that $$K(G, 1)=BG$$, so for $$i=1$$, those cohomology operations are in $$H^2(BG, H)$$, and therefore given by the Bocksteins of the corresponding central extensions of $$G$$ by $$H$$. Also, for $$G=H=\mathbb{Z}_p$$, the stable cohomology operations are given by the Steenrod algebra, and the only degree-1 elements are Bocksteins.

However, I don't know how it is for unstable cohomology operations, or groups other than $$\mathbb{Z}_p$$. I'm mostly interested in simple groups, such as finitely generated or $$\mathbb{R}/\mathbb{Z}$$.

• We can make $K(G,i)$ by gluing to a point one $i$-cell for each generator of $G$, then one $i+1$-cell for each relation, and then additional cells to kill the remaining homotopy groups. The remaining homotopy groups are in degree $i+1$ and above, so we kill them by adding cells of dimension $i+2$ and above, which can't add any new cohomology classes in degree $i+1$. Then the cells we added in degree $i+1$ for the relations must give the Bocksteins. Jan 28 at 20:46

Yes. For $$i\ge1$$ you can build $$K(G,i)$$ from the Moore space $$M(G,i)$$ by adding cells of dimension $$\ge i+2$$, so $$H_i(K(G,i); Z) = G$$ and $$H_{i+1}(K(G,i); Z) = 0$$. Hence $$Ext(G, H) \cong H^{i+1}(K(G,i); H)$$ by the UCT. The elements of $$H^{i+1}(K(G,i); H)$$ represent the cohomology operations $$H^i(-;G) \to H^{i+1}(-;H)$$, and the elements of $$Ext(G,H)$$ correspond to the Bockstein operations. The case $$i=0$$ may require special attention.