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Crossed modules and homology. (English) Zbl 0807.18008

The first example of crossed modules was produced by J. H. C. Whitehead [Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.388)] for the study of relative homotopy groups: if \(Y \subset X\), \(\pi_ 2 (X,Y)\) is a crossed \(\pi_ 1 (Y)\)-module.
Denote by \((T,G, \partial)\), \(\partial : T \to G\), a crossed \(G\)-module. The authors define the two homology groups \(H_ 1(T,G, \partial)\) and \(H_ 2 (T,G, \partial)\) of \((T,G, \partial)\). For instance, seeing a group \(G\) as a crossed module in the two usual ways, one has: \(H_ k (1,G,i) = (1,H_ k (G),i)\) and \(H_ k (G,G, \text{id}) = (H_ k (G), H_ k (G), \text{id})\), \(k = 1,2\). For any short exact sequence of crossed modules (with a suitable hypothesis), these homology groups fit together in a (natural) five-term exact sequence of crossed modules.

MSC:

18G30 Simplicial sets; simplicial objects in a category (MSC2010)
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology

Citations:

Zbl 0040.388
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References:

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