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