  We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma'$
immering into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to
\Gamma$, we consider the map $\Lambda'$ from the circular components
$\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the
degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the
Euler characteristic of $\Gamma'$. As a consequence, we obtain a homological
version of coherence for one-relator groups.

Group Theory