The surgery exact triangle in $\mathrm{Pin}(2)\mskip-1.5mu$–monopole Floer homology

Abstract

We prove the existence of an exact triangle for the $Pin\left(2\right)$–monopole Floer homology groups of three-manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this triangle to describe the Manolescu correction terms of the manifolds obtained by $\left(\pm 1\right)$–surgery on alternating knots with Arf invariant $1$.