The surgery exact triangle in Pin(2)-monopole Floer homology

TL;DR

This paper establishes an exact triangle in Pin(2)-monopole Floer homology for three manifolds related by Dehn surgeries, revealing new relations and invariants for homology spheres from alternating knots.

Contribution

It proves a new exact triangle in Pin(2)-monopole Floer homology and applies it to compute invariants of homology spheres from alternating knots.

Findings

Established the exact triangle for Pin(2)-monopole Floer homology.

Described invariants for homology spheres from ( extpm 1)-surgery on alternating knots.

Identified differences from the usual monopole Floer homology triangle.

Abstract

We prove the existence of an exact triangle for the Pin(2)-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 invariants associated to homology spheres obtained by (\pm1)-surgery on alternating knots.