Spin structures and codimension-two homeomorphism extensions

Let $\mathrm{ı}:M\to {\text{RR}}^{p+2}$ be a smooth embedding from a connected, oriented, closed $p$-dimesional smooth manifold to ${\text{RR}}^{p+2}$, then there is a spin structure ${\mathrm{ı}}^{\mathrm{♯}}({\varsigma }^{p+2})$ on $M$ canonically induced from the embedding. If an orientation-preserving diffeomorphism $\tau$ of $M$ extends over $\mathrm{ı}$ as an orientation-preserving topological homeomorphism of ${\text{RR}}^{p+2}$, then $\tau$ preserves the induced spin structure. Let ${\text{esg}}_{\text{cat}}(\mathrm{ı})$ be the subgroup of the $\text{cat}$-mapping class group ${\text{mcg}}_{\text{cat}}(M)$ consisting of elements whose representatives extend over ${\text{RR}}^{p+2}$ as orientation-preserving $\text{cat}$-homeomorphisms, where $\text{cat}=\text{topo}$, $\text{pl}$ or $\text{diff}$. The invariance of ${\mathrm{ı}}^{\mathrm{♯}}({\varsigma }^{p+2})$ gives nontrivial lower bounds to $[{\text{mcg}}_{\text{cat}}(M):{\text{esg}}_{\text{cat}}(\mathrm{ı})]$ in various special cases. We apply this to embedded surfaces in ${\text{RR}}^4$ and embedded $p$-dimensional tori in ${\text{RR}}^{p+2}$. In particular, in these cases the index lower bounds for ${\text{esg}}_{\text{topo}}(\mathrm{ı})$ are achieved for unknotted embeddings.