Computability of Homology groups

Abstract. Some notes about the computability of homology groups of chain complexes. These notes are based on [3,6,5,7,1,4,2]. 1 Group theory Theorem 1 (First Isomorphism Theorem). Let G and H be groups, and let ϕ: G → H be a surjective homomorphism, then H ' G / kerϕ. The above theorem corresponds with first_isog SSReflect theorem. Theorem 2. Let G be a group and N E G. Then, the map pi: G → G/N g 7 → gN is a surjective homomorphism. Moreover, kerpi = N. The above theorem corresponds with ker_coset SSReflect theorem. Definition 1. The direct product of several groups G1,..., Gn is the Cartesian product endowed with an operation defined elementwise. The direct sum of the groups G1,..., Gn is denoted by G1 ×...×Gn. When the groups involved are abelian and written with additive notation, it is common to use the terminology direct sum instead of direct product and use the notation G⊕H instead of G×H. Lemma 1 (Lemma 3.6.1 of [3]). If Hi is a subgroup of an Abelian group Gi, the