Monopole triangle over integers

We prove the surgery exact triangle for monopole (Seiberg–Witten) Floer homology over integer coefficients, extending the work of Kronheimer–Mrowka–Ozsváth–Szabó over $\mathbb{Z}\mathrm{/}2$, Lin–Ruberman–Saveliev over $\mathbb{Q}$, and Freeman over $\mathbb{Z}[\sqrt{-1}]$. Our proof is based on a modification of Kronheimer–Mrowka’s local system on monopole Floer homology and an adaptation of Freeman’s computation. As a standard application, following Bloom and Scaduto, we obtain a spectral sequence ${\widetilde{Kh}}_{\mathrm{o}\mathrm{d}\mathrm{d}}(L)\Rightarrow {\widetilde{HM}}_{\bullet }(-{\mathrm{\Sigma }}_2(L))$ over integer coefficients for an oriented link $L\subset S^3$, thereby solving Ozsváth–Rasmussen–Szabó’s conjecture.