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.
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