  In complex dynamics, the boundaries of higher dimensional hyperbolic
components in holomorphic families of polynomials or rational maps are
mysterious objects, whose topological and analytic properties are fundamental
problems.
  In this paper, we show that in some typical families of polynomials (i.e.
algebraic varieties defined by periodic critical relations), the boundary of a
capture hyperbolic component $\mathcal H$ is homeomorphic to the sphere
$S^{2\dim_\mathbb{C}(\mathcal{H})-1}$. Furthermore, we establish an unexpected
identity for the Hausdorff dimension of $\partial \mathcal H$:
$$\operatorname{H{.}dim}(\partial\mathcal{H}) = 2
\dim_\mathbb{C}(\mathcal{H})-2+\max_{f\in\partial\mathcal{H}}
\operatorname{H{.}dim}(\partial A^J(f)),$$ where $A^J(f)$ is the union of the
bounded attracting Fatou components of $f$ associated with the free critical
points in the Julia set $J(f)$.
  In the proof, some new results with independent interests are discovered.
