2022/10/25 by Aliakbar Daemi, Daemi, Aliakbar, Mike Miller Eismeier +1
Mathematics · Medicine · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Botulinum Toxin and Related Neurological Disorders
paper · pdf · doi:10.48550/arxiv.2210.14071
In previous work, the second author defined 'equivariant instanton homology groups' I^\bullet(Y,π;R) for a rational homology 3-sphere Y, a set of auxiliary data π, and a PID R. These objects are modules over the cohomology ring H-*(BSO3;R). We prove that the equivariant instanton homology groups I^\bullet(Y;R) are independent of the auxiliary data π, 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, π; R) which now depends on the auxiliary data π, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula'' for I_*(Y, π; R) as the auxiliary data π moves between adjacent chambers. We use this to define an instanton invariant λI(Y) ∈ \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 → 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'.