In the previous work, the second authors used the rational homology 3 category $Y$, the set of ancillary data $\pi$, and the PID $R$. These objects are modules on the cohomology ring $H^{-*}(BSO_3;R)$.
Prove that the equivariant instanton homology group $I^\bullet(Y;R)$ is independent of the auxiliary data $\pi$ and define the topological invariant of the rational homology sphere. Furthermore, we prove that these invariants are functors under cobordism of 3-manifolds with paths between boundary components.
For any rational homology sphere $Y$, we can also define an analogue of the irreducible instanton homology group of Floer’s integer homology sphere $I_*(Y, \pi; R)$. Equivariant instanton homology group. However, when the ancillary data $\pi$ moves between adjacent chambers, our method allows us to prove the exact “wall-crossing formula” for $I_*(Y, \pi; R)$. increase. Use this to define an instanton. Invariant $\lambda_I(Y) \in \Bbb Q$ of the rational homology sphere, which is a priori equivalent to the Casson-Walker invariant.
Our approach to immutability uses a new technique known as the suspended flow category. Obstructed coboldisms supporting reducible instantons that cannot be cut laterally or removed by small changes in perturbation. Eliminates and replaces neighbors of the solution with Instanton. Do not define chain maps between instanton chain complexes of $Y$ and $Y’$, because the resulting moduli space has a new type of boundary component. However, it defines a chain map between the instanton chain complex at $Y$ and a kind of suspension of the instanton chain complex at $Y’$.
