Instantons and rational homology spheres

TL;DR

This paper develops topological invariants for rational homology spheres using equivariant instanton homology, proves their independence from auxiliary data, and introduces a novel technique called suspended flow category for invariance proofs.

Contribution

It establishes the invariance of equivariant instanton homology groups for rational homology spheres and introduces a suspended flow category technique for handling obstructed cobordisms.

Findings

Proved independence of equivariant instanton homology from auxiliary data.

Defined an instanton invariant conjecturally equal to the Casson-Walker invariant.

Developed a suspended flow category method for invariance under obstructed cobordisms.

Abstract

In previous work, the second author defined 'equivariant instanton homology groups' $I^\bullet(Y,\pi;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $\pi$, and a PID $R$. These objects are modules over the cohomology ring $H^{-*}(BSO_3;R)$. We prove that the equivariant instanton homology groups $I^\bullet(Y;R)$ are independent of the auxiliary data $\pi$, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components. For any rational homology sphere $Y$, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres $I_*(Y, \pi; R)$ which now depends on the auxiliary data $\pi$, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula'' for $I_*(Y, \pi; R)$ as the auxiliary data $\pi$ moves between adjacent chambers. We use this to define an instanton invariant $\lambda_I(Y) \in \Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant. Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism $W: Y \to Y'$, which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of $Y$ and $Y'$. However, it does define a chain map between the instanton chain complex of $Y$ and a sort of suspension of the instanton chain complex of $Y'$.