Calculating the homology of Poincaré's homology sphere through dodecahedron and cellular homology
Homology sphere A homology sphere is a (closed connected oriented) $n$-mainfolds with the same homology as the $n$-sphere. Poincaré first conjectured that any $n$-manifolds homologous (i.e. having the same homology) to the $n$-sphere should be homeomorphic to the $n$-sphere, then later he found a counterexample, which led him to a modified conjecture (which is the Poincaré’s conjecture). The counterexample he found is very interesting, one of the way to construct it is the folowing:...