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

Projective Freeness of Algebras of Bounded Holomorphic Functions on Infinitely Connected Domains