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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Application of group cohomology to space constructions


Authors: Paul Igodt and Kyung Bai Lee
Journal: Trans. Amer. Math. Soc. 304 (1987), 69-82
MSC: Primary 57S30; Secondary 20J10
MathSciNet review: 906806
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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)\mathop \to ... ...(G;\,K):{\psi _{{\text{out}}}} = {\phi _{{\text{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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0906806-2
PII: S 0002-9947(1987)0906806-2
Keywords: Nonabelian group cohomology, Seifert fiber spaces, infranil-manifolds
Article copyright: © Copyright 1987 American Mathematical Society