of genus $g$ and $\mathcal {M}_{g}$ be the mapping class group

of $\Sigma_{g}$; i.e.,the group of all isotopy classes of

orientation-preserving self-diffeomorphisms of $\Sigma_{g}$.

We define $SP_{g}[q]$ as the subgroup of $\mathcal {M}_{g}$

consisting of mapping classes which preserve the given spin

structure associated to the quadratic form $q$ on

$H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$. \\

As is well-known, the automorphisms over

$H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$

form the $\mathbb{Z}_{2}$-symplectic group

$Sp(2g; \mathbb{Z}_{2})$. In this talk, we will observe

automorphisms over $H_{1}(\Sigma_{g} ; \mathbb{Z}_{2})$

induced by the elements of $SP_{g}[q]$,

which form the subgroup of $Sp(2g; \mathbb{Z}_{2})$.

We will call this group

the spin-preserving symplectic group, and

determine it explicitly where the case is $g=1$ and $g=2$.

respectivily.

We call the images of surfaces embedded in the $4$-shpere

surface-knots. Lastly we give an application to the

surface-knot of genus two.