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Cohomology of nilmanifolds and torsion-free, nilpotent groups


Authors: Larry A. Lambe and Stewart B. Priddy
Journal: Trans. Amer. Math. Soc. 273 (1982), 39-55
MSC: Primary 57T15; Secondary 17B56, 22E25, 58A12
DOI: https://doi.org/10.1090/S0002-9947-1982-0664028-2
MathSciNet review: 664028
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Abstract: Let $ M$ be a nilmanifold, i.e. $ M = G/D$ where $ G$ is a simply connected, nilpotent Lie group and $ D$ is a discrete uniform, nilpotent subgroup. Then $ M \simeq K(D,1)$. Now $ D$ has the structure of an algebraic group and so has an associated algebraic group Lie algebra $ L(D)$. The integral cohomology of $ M$ is shown to be isomorphic to the Lie algebra cohomology of $ L(D)$ except for some small primes depending on $ D$. This gives an effective procedure for computing the cohomology of $ M$ and therefore the group cohomology of $ D$. The proof uses a version of form cohomology defined for subrings of $ {\mathbf{Q}}$ and a type of Hirsch Lemma. Examples, including the important unipotent case, are also discussed.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0664028-2
Keywords: Nilmanifolds, torsion-free, nilpotent groups, formal groups, de Rham cohomology
Article copyright: © Copyright 1982 American Mathematical Society

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