Application of group cohomology to space constructions

Abstract:

From a short exact sequence of crossed modules $1 \to K \to H \to \bar H \to 1$ and a $2$-cocycle $(\phi ,\,h) \in {Z^2}(G;\,H)$, a $4$-term cohomology exact sequence $H_{ab}^1(G;Z) \to H_{(\bar {\phi }, \bar {h})}^1 (G; \bar {H}, \bar {Z}) \stackrel {\delta }{\to } \bigcup \{ H_\psi ^2 (G;K) : \psi _{\mathrm {out}} = \phi _{\mathrm {out}} \} \to H_{ab}^2(G;\,Z)$ is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and $Z$ is the center of the crossed module $H$. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.