Determine all cohomology operations \(H^1(X; \mathbb{Z}) \to H^n(X; \mathbb{Z})\), \(H^2(X; \mathbb{Z}) \to H^n(X; \mathbb{Z})\), and \(H^1(X; \mathbb{Z}_p) \to H^n(X; \mathbb{Z}_p)\) for \(p\) prime.
Solution
Cohomology operations \[ H^m(-;G)\longrightarrow H^n(-;H) \] are represented by classes in \[ H^n(K(G,m);H). \] We apply this to the three cases.
For \(H^1(-;\mathbb Z)\), the representing space is \[ K(\mathbb Z,1)=S^1. \] Hence \[ H^n(S^1;\mathbb Z)= \begin{cases} \mathbb Z,&n=1,\\ 0,&n>1. \end{cases} \] Thus the only positive-degree operations are \[ \boxed{u\longmapsto a\,u\qquad(a\in\mathbb Z)} \] in degree \(1\); all operations to \(H^n\) for \(n>1\) are zero.
For \(H^2(-;\mathbb Z)\), use \[ K(\mathbb Z,2)=\mathbb{CP}^\infty, \qquad H^*(\mathbb{CP}^\infty;\mathbb Z)=\mathbb Z[x],\quad |x|=2. \] Therefore \[ H^n(K(\mathbb Z,2);\mathbb Z)= \begin{cases} \mathbb Z\,x^j,&n=2j,\\ 0,&n\text{ odd}. \end{cases} \] Hence every operation is \[ \boxed{u\longmapsto a\,u^j\quad(n=2j),\qquad a\in\mathbb Z,} \] and there are no nonzero operations in odd target degree.
Finally consider \(H^1(-;\mathbb Z_p)\). The representing space is \(K(\mathbb Z_p,1)\). For \(p=2\), \[ H^*(K(\mathbb Z_2,1);\mathbb Z_2)=\mathbb Z_2[x], \qquad |x|=1. \] Thus in every degree \(n>0\) the group of operations is one-dimensional, generated by \[ \boxed{u\longmapsto u^n.} \]
For odd \(p\), \[ H^*(K(\mathbb Z_p,1);\mathbb Z_p) =\Lambda(x)\otimes\mathbb Z_p[y], \qquad |x|=1,\quad |y|=2,\quad y=\beta x, \] where \(\beta\) is the mod-\(p\) Bockstein. Thus there is again one basis operation in every positive degree: \[ \boxed{ \begin{aligned} H^1(-;\mathbb Z_p)&\to H^{2j}(-;\mathbb Z_p), &u&\mapsto (\beta u)^j,\\ H^1(-;\mathbb Z_p)&\to H^{2j+1}(-;\mathbb Z_p), &u&\mapsto u(\beta u)^j. \end{aligned}} \] All operations are scalar multiples of these by elements of \(\mathbb Z_p\).