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Proceedings of the American Mathematical Society
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
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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 $Z_\ast(G)$.


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Additional Information

Goutam Mukherjee
Affiliation: Stat-Math Division, Indian Statistical Institute, 203 B. T. Road, Calcutta-700 035, India
Email: goutam@isical.ernet.in

Parameswaran Sankaran
Affiliation: SPIC Mathematical Institute, 92 G. N. Chetty Road, Madras-600 017, India
Email: sankaran@smi.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04133-1
PII: S 0002-9939(98)04133-1
Received by editor(s): July 11, 1996
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1998 American Mathematical Society