Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 664028
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [C] H. Cartan, Théories cohomologiques, Invent. Math. 35 (1976), 261-271. MR 0431137 (55:4139)
  • [CE] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • [CP] B. Cenkl and R. Porter, De Rham Theorem with cubical forms (preprint). MR 739139 (85j:55009)
  • [Cv] C. Chevalley, Theory of Lie groups, Princeton Univ. Press, Princeton, N. J., 1946.
  • [G] W. Grueb, Multilinear algebra, Universitext, Springer-Verlag, New York, 1978. MR 504976 (80c:15017)
  • [H] P. Hall, Nilpotent groups, Canad. Math. Congress, Edmonton, 1957 (reissued by Queens College, London).
  • [Hi] G. Hirsch, L'anneau de cohomologie d'un espace fibré en spheres, C. R. Acad. Sci. Paris Sér. A-B 241(1955), 1021-1023. MR 0077120 (17:993d)
  • [Hu] J. E. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975. MR 0396773 (53:633)
  • [Lz] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Ecole Norm. Sup. 71 (1954), 101-190. MR 0088496 (19:529b)
  • [Mc] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Springer-Verlag, New York, 1963.
  • [MB] S. Mac Lane and G. Birkhoff, Algebra, Macmillan, New York, 1971.
  • [M] A. Malcev, On a class of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 9-32; English transl., Math. USSR-Izv. 39 (1949). MR 0028842 (10:507d)
  • [Mi] E. Miller, DeRham cohomology with arbitrary coefficients, Topology 17 (1978), 193-203. MR 0467722 (57:7575)
  • [N] K. Nomizu, On the cohomology of compact homogeneous space of nilpotent Lie groups, Ann. of Math. (2)59 (1954), 531-538. MR 0064057 (16:219c)
  • [Q] D. Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968). MR 0231919 (38:245)
  • [S] J. P. Serre, Lie algebras and Lie groups, Benjamin, New York, 1965. MR 0218496 (36:1582)
  • [St] R. Steinberg, Lectures on Chevally groups, Yale University Lecture Notes, 1967. MR 0466335 (57:6215)
  • [ST] H. Shulman and D. Tishler, Leaf invariants for foliations and the Van Est isomorphism, J. Differential Geom. 11 (1976), 539-546. MR 0451262 (56:9549)
  • [Su] D. Sullivan, Infinitesimal computations in topology, IHES Publ. Math. 47 (1977), 269-331. MR 0646078 (58:31119)
  • [V] V. Varadarajan, Lie groups, Lie algebras and their representations, Prentice-Hall, Englewood Cliffs, N. J., 1974. MR 0376938 (51:13113)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57T15, 17B56, 22E25, 58A12

Retrieve articles in all journals with MSC: 57T15, 17B56, 22E25, 58A12

Additional Information

Keywords: Nilmanifolds, torsion-free, nilpotent groups, formal groups, de Rham cohomology
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society