Topology of the Maximal Ideal Space of $H^{\mathrm{\infty }}$ Revisited

Let $M(H^{\mathrm{\infty }})$ be the maximal ideal space of the Banach algebra $H^{\mathrm{\infty }}$ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$. We prove that $M(H^{\mathrm{\infty }})$ is homeomorphic to the Freudenthal compactification $\gamma (M_a)$ of the set $M_a$ of all non-trivial (analytic disks) Gleason parts of $M(H^{\mathrm{\infty }})$. Also, we give alternative proofs of important results of Su'{a}rez asserting that the set $M_s$ of trivial (one-pointed) Gleason parts of $M(H^{\mathrm{\infty }})$ is totally disconnected and that the \v{C}ech cohomology group $H^2(M(H^{\mathrm{\infty }}),\mathbb{Z})=0$.