Topology of the maximal ideal space of H∞ revisited

Let M(H∞) be the maximal ideal space of the Banach algebra H∞ of bounded holomorphic functions on the unit disk D⊂C. We prove that M(H∞) is homeomorphic to the Freudenthal compactification γ(Ma) of the set Ma of all non-trivial (analytic disks) Gleason parts of M(H∞). Also, we give alternative proofs of important results of Suárez asserting that the set Ms of trivial (one-pointed) Gleason parts of M(H∞) is totally disconnected and that the Čech cohomology group H2(M(H∞),Z)=0.