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On the second cohomology group of a simplicial group. (English) Zbl 1203.55011
Eilenberg and Mac Lane have shown that the second cohomology groups of path-connected topological spaces can be expressed in terms of their first two homotopy groups and their first Postnikov invariants. In this paper the author proves the analogous result for simplicial groups, using the theory of crossed module extensions of groups. Since simplicial groups form a model for path-connected spaces, this give a more algebraic proof for the result of Eilenberg and Mac Lane.
##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 20J06 Cohomology of groups 55S45 Postnikov systems, $$k$$-invariants
##### Keywords:
simplicial group; crossed module
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