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

Let $M(H^\infty)$ be the maximal ideal space of the Banach algebra $H^\infty$ of bounded holomorphic functions on the unit disk $\mathbb D\subset\mathbb C$. We prove that $M(H^\infty)$ is homeomorphic to the Freudenthal compactification $\gamma(M_a)$ of the set $M_a$ of all non-trivial (analytic disks) Gleason parts of $M(H^\infty)$. 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 $M(H^\infty)$ is totally disconnected and that the \v{C}ech cohomology group $H^2(M(H^\infty),\mathbb Z)=0$.