Relation To Topology
In topology, groups can often be described as finitely presented groups and a fundamental question is to calculate their integral homology Hn(G, Z). In particular, the second homology plays a special role and this led Hopf to find an effective method for calculating it. The method in (Hopf 1942) is also known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:
where G ≅ F/R and F is a free group. The same formula also holds when G is a perfect group.
The recognition that these formulas were the same led Eilenberg and Mac Lane to the creation of cohomology of groups. In general,
where the star denotes the algebraic dual group. Moreover when G is finite, there is an unnatural isomorphism
The Hopf formula for H2(G) has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below.
A perfect group is one whose first integral homology vanishes. A superperfect group is one whose first two integral homology groups vanish. The Schur covers of finite perfect groups are superperfect. An acyclic group is a group all of whose reduced integral homology vanishes.
Read more about this topic: Schur Multiplier
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