Surgery Exact Triangles in Instanton Theory

Abstract

We prove an exact triangle relating knot instanton Floer homology to the instanton homology of surgeries along the knot. To the author's knowledge, this is the first such result in instanton homology with integer coefficients and has no analogue in Heegaard Floer homology. To illustrate the latter claim, we derive as a consequence of this triangle, building on previous computations in the literature, that the Poincar\'e Homology Sphere is not an instanton $L$-space with $\mathbb{Z}/2$-coefficients.