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Cameron Gordon's theorem that H_2(G) $H_2(G)$ is generally non-computable from a group presentation of G. IMO this should be the standard appended caveat to Hopf's formula.

Gordon, C. Some embedding theorems and undecidability questions for groups. Combinatorial and geometric group theory (Edinburgh, 1993), 105--110, London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995.

2 ref

Cameron Gordon's theorem that H_2(G) is generally non-computable from a group presentation of G. IMO this should be the standard appended caveat to Hopf's formula.

Gordon, C. Some embedding theorems and undecidability questions for groups. Combinatorial and geometric group theory (Edinburgh, 1993), 105--110, London Math. Soc. Lecture Note Ser., 204, Cambridge Univ. Press, Cambridge, 1995.