Monopole triangle over integers

Abstract

We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsváth--Szabó over $\mathbb{Z}/2$, 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 $\widetilde{Kh}_{\mathrm{odd}}(L)\Rightarrow \widetilde{HM}_\bullet(-Σ_2(L))$ over integer coefficients for an oriented link $L\subset S^3$, thereby solving Ozsváth--Rasmussen--Szabó's conjecture.