Is there any simple example that $lim^1$ terms appear?

Question

limit of cohomology does not behave well in the sense that there will be $lim^1$ term. Is there any simple example that $lim^1$ terms appear?

Thanks!

Answer 1

Consider the tower of spaces (I take $S^1 = K(\mathbb{Z}, 1)$ for simplicity, but any $K(\mathbb{Z}, n)$ would do): $$\ldots \to S^1 \xrightarrow{\cdot^p} S^1 \xrightarrow{\cdot^p} S^1$$

Where I'm considering $X_s = S^1 = \{ z \in \mathbb{C} : |z| = 1 \}$, $p$ is a prime number, and $X_s \to X_{s-1}$ is raising to the $p$th power (you may have to replace these by fibrations). Let $X = \varprojlim X_s$. Then the Milnor exact sequence (this is how it's called in Goerss and Jardine' Simplicial Homotopy Theory -- this example is basically example VI.2.18 in there) tells you that the following exact sequence is exact (everything is abelian): $$0 \to \varprojlim_s{}^1 \pi_{i+1} X_s \to \pi_i X \to \varprojlim_s \pi_i X_s \to 0$$

In particular for $i = 0$, you find that $\pi_0 X = \varprojlim_s{}^1 p^s \mathbb{Z} \cong \mathbb{Z}_p / \mathbb{Z}$ (where $\mathbb{Z}_p$ are the $p$-adic integers). Going to $H^0$ you see that $H^0(X) \cong \mathbb{Z}_p / \mathbb{Z}$.