Stackings and the W-cycle conjecture

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

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