Question

I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space):

$$H^{n+1}(K(\mathbb Z_p,n);\mathbb Z_p) \cong \mathbb Z_p.$$

(This would be obvious if n+1 were replaced with n. This is supposed to imply that the natural transformations $H^n(X; \mathbb Z_p)\to H^{n+1}(X; \mathbb Z_p)$ are all multiples of the Bockstein homomorphism).

I'm at a loss trying to understand why. Spectral sequences haven't been covered yet, so there should be some simple reason. Also, is there a way to see the Bockstein in all this?

Thank you!

Answer 1

Universal coefficient theorem + $H_{n+1}(K(Z_p,n);Z)=0$. An elementary way to see the latter is that the single $n+1$ cell added to $S^n$ to kill $p$ times the generator is not a cellular cycle.