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Some computations in equivariant cobordism in relation to Milnor manifolds. (English) Zbl 1295.57036
The Milnor manifolds \({\mathcal H}(m,n)\), \(m\leq n\), are defined by \[ {\mathcal{H}}(m,n)=\{([x_0:\cdots:x_m],[y_0:\cdots:y_n])\mid \sum_{j=0}^m x_jy_j=0 \}\subset {\mathbb R}P^m\times {\mathbb R}P^n. \] In this paper the authors consider the natural action of the \(2\)-torus \(G={\mathbb Z}_2^n\) on \({\mathcal H}(m,n)\) and show, using a criterion proven in [G. Mukherjee and P. Sankaran, Proc. Am. Math. Soc. 126, No. 2, 595–606 (1998; Zbl 0889.57041)], that it often defines indecomposable elements in the equivariant cobordism ring \(Z_*(G)\) of smooth closed manifolds with a \(G\)-action having finite fixed point set. On the other hand, forgetting the group action, it is known which Milnor manifolds are boundaries [S. S. Khare and A. K. Das, Indian J. Pure Appl. Math. 31, No. 11, 1503–1513 (2000; Zbl 0967.57029)], thus the result provides information on the kernel of the forgetful homomorphism \(Z_*(G)\to {\mathcal N}_*\) to the ordinary unoriented cobordism ring.

MSC:
57R85 Equivariant cobordism
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57S17 Finite transformation groups
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