Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Elementary abelian 2-group actions
on flag manifolds and applications

Authors: Goutam Mukherjee and Parameswaran Sankaran
Journal: Proc. Amer. Math. Soc. 126 (1998), 595-606
MSC (1991): Primary 57R75, 57R85
MathSciNet review: 1423325
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal N_\ast$ denote the unoriented cobordism ring. Let $G=(\mathbb Z/2)^n$ and let $Z_\ast(G)$ denote the equivariant cobordism ring of smooth manifolds with smooth $G$-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold $M=O(m)/(O(m_1)\times\dots\times O(m_s))$ is in the subalgebra generated by $\bigoplus _{i<2^n}\mathcal N_i$, where $m= \sum m_j$, and $2^{n-1}<m\le 2^n$. We obtain sufficient conditions for indecomposability of an element in $Z_\ast(G)$. We also obtain a sufficient condition for algebraic independence of any set of elements in $Z_\ast(G)$. Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map $Z_d(G)\to\mathcal N_d$ in dimensions $2\le d<n$, for $n>2$, and show that they generate a polynomial subalgebra of $Z_\ast(G)$.

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

  • 1. P. E. Conner, Differentiable periodic maps, 2nd Ed., L.N.M. (738) Springer-Verlag, 1979. MR 81f:57018
  • 2. P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse Sr-33, Springer-Verlag, 1964. MR 31:750
  • 3. T. tom Dieck, Fixpunkte vertauschbarer involutionen, Archiv der math., 20, 295-298, 1969. MR 42:3798
  • 4. -, Characteristic numbers of $G$ manifolds. I, Invent. Math. 13, 213-224, 1971. MR 46:8236
  • 5. C. Kosniowski and R. E. Stong, $(\mathbb Z_2)^k$-Actions and Characteristic numbers, Indiana Univ. Math. J., 28, 725-743, 1979. MR 81d:57027
  • 6. K. Y. Lam, A formula for the tangent bundle of flag manifolds and related manifolds, Trans. Amer. Math. Soc., 213, 305-314, 1975. MR 55:4196
  • 7. J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. Math. Stud., 76, Princeton, 1974. MR 55:13428
  • 8. G. Mukherjee, Equivariant cobordism of Grassmann and flag manifolds, Proc. Ind. Acad. Sci., 105, 381-391, 1995. MR 97g:57045
  • 9. P. Sankaran, Determination of Grassmann manifolds which are boundaries, Canad. Math. Bull., 34, 119-122, 1991. MR 92h:57049
  • 10. P. Sankaran and K. Varadarajan, Group actions on flag manifolds and cobordism, Canad. J. Math., 45, 650-661, 1993. MR 94k:57050
  • 11. R. E. Stong, Equivariant bordism and $(\mathbb Z_2)^k$-actions, Duke Math. J. 37, 779-785, 1970. MR 42:6847
  • 12. R. E. Stong (reviewer). MR 89d:57050

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57R75, 57R85

Retrieve articles in all journals with MSC (1991): 57R75, 57R85

Additional Information

Goutam Mukherjee
Affiliation: Stat-Math Division, Indian Statistical Institute, 203 B. T. Road, Calcutta-700 035, India

Parameswaran Sankaran
Affiliation: SPIC Mathematical Institute, 92 G. N. Chetty Road, Madras-600 017, India

Received by editor(s): July 11, 1996
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society