  The algebra $H^\infty(D)$ of bounded holomorphic functions on
$D\subset\mathbb C$ is projective free for a wide class of infinitely connected
domains. In particular, for such $D$ every rectangular left-invertible matrix
with entries in $H^\infty(D)$ can be extended in this class of matrices to an
invertible square matrix (the generalization of the corona theorem for
$H^\infty(D)$). This follows from a new result on the structure of the maximal
ideal space of $H^\infty(D)$ asserting that its covering dimension is $2$ and
the second \v{C}ech cohomology group is trivial.
