  On the Sierpinski gasket $\mathcal{SG}$, we consider Sobolev spaces
$L^2_\sigma(\mathcal{SG})$ associated with the standard Laplacian $\Delta$ with
order $\sigma\geq 0$. When $\sigma\in\mathbb{Z}^+$, $L^2_\sigma(\mathcal{SG})$
consists of functions equipped with $L^2$ norms of the function itself and its
Laplacians up to $\sigma$ order; when $\sigma\notin \mathbb{Z}^+$, we fill up
the gaps between integer orders by using complex interpolation. Let
$L^2_{\sigma, D}(\mathcal{SG})=(I-\Delta_D)^{-\sigma}L^2(\mathcal{SG})$ where
$\Delta_D$ is the Dirichlet Laplacian associated with $\Delta$. Let
$\{p_n\}_{n\geq 0}$ be a collection of countably many points located along one
of the symmetrical axes of $\mathcal{SG}$. We make a full characterization of
the trace spaces of $L^2_\sigma(\mathcal{SG})$ and
$L^2_{\sigma,D}(\mathcal{SG})$ to $\{p_n\}_{n\geq 0}$. Using this, we get a
full description of the relationship between $L^2_\sigma(\mathcal{SG})$ and
$L^2_{\sigma,D}(\mathcal{SG})$ for $\sigma\geq 0$. The result indicates that
when $\sigma-\frac{\log 3}{2\log5}\in \mathbb{Z}^+$, $L^2_{\sigma,
D}(\mathcal{SG})$ is not closed in $L^2_{\sigma}(\mathcal{SG})$ and has an
infinite codimension. Otherwise, $L^2_{\sigma, D}(\mathcal{SG})$ is closed in
$L^2_{\sigma}(\mathcal{SG})$ with a finite codimension. Similar result holds
for the Neumann case. Another consequence of the trace result is that the
Sobolev spaces $L^2_\sigma(\mathcal{SG})$ are stable under complex
interpolation for $\sigma\geq 0$ although they are defined by piecewise
interpolation between integer orders.
