Characterization of Y-equivalence for bordered spin 3-manifolds

Let $(M,\phi,s)$ and $(M',\phi',s')$ be $\mathbb{S}$-bordered spin $3$-manifolds. Define

$$R_8((M,\phi,s),(M',\phi',s'))=(R(M”,s”)\bmod 8)\in\mathbb{Z}_8,$$

where $M”=(-M)\cup_{\phi,\phi'}M'$ and $s”\in\operatorname{Spin}(M”)$ is any gluing of $s$ and $s'$. A homology isomorphism from $(M,\phi)$ to $(M',\phi')$ means an isomorphism preserving the relevant bordered homology data.

Main conjecture. The following conditions are equivalent: $(M,\phi,s)$ and $(M',\phi',s')$ are $Y$-equivalent; and there is a homology isomorphism from $(M,\phi)$ to $(M',\phi')$ such that

$$R_8((M,\phi,s),(M',\phi',s'))=0\pmod {8}.$$

The conjecture proposes that homology data together with the Rochlin invariant modulo $8$ completely characterizes $Y$-equivalence of $\mathbb{S}$-bordered spin $3$-manifolds. The preceding lemma shows that $R_8$ depends only on the $Y$-equivalence classes, while the paper's main theorem reduces the characterization to the corresponding unbordered equivalence statement.