Hatcher’s Algebraic Topology unfortunately hadn’t been written when I studied the subject in grad school, but a few years ago I participated in a meetup that went through about half of it. I wrote up notes on things I puzzled over, or just observations I found helpful. Here they are. Besides the homology group mentioned in the title, some other tidbits: a figure to elucidate the calculation of the fundamental group of the complement of the Alexander Horned Sphere, and more details for the intuitive proof Hatcher sketches of PoincarĂ© duality.
Diagonal Argument 