Exact sequence giving kernel of mapping from tensor square to exterior square

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Statement

Let G be any group. There is a natural exact sequence as follows:

H_3(G;\mathbb{Z}) \to \Gamma(G^{\operatorname{ab}}) \to G \otimes G \to G \wedge G \to 0

Here:

The maps are as follows:

  • The first map is mysterious (some kind of connecting homomorphism?)
  • The second map, \Gamma(G^{\operatorname{ab}}) \to G \otimes G, sends PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
  • The third map, G \otimes G \to G \wedge G, is the natural quotient map that sends a generator of the form x \otimes y to the corresponding element x \wedge y.

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