Basu, Samik; Mukherjee, Goutam; Sarkar, Swagata Some computations in equivariant cobordism in relation to Milnor manifolds. (English) Zbl 1295.57036 Topology Appl. 172, 1-9 (2014). 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. Reviewer: Oliver Goertsches (Hamburg) Cited in 1 Document MSC: 57R85 Equivariant cobordism 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 57S17 Finite transformation groups Keywords:bordism; 2-torus action PDF BibTeX XML Cite \textit{S. Basu} et al., Topology Appl. 172, 1--9 (2014; Zbl 1295.57036) Full Text: DOI arXiv References: [1] Conner, P. E., Differentiable periodic maps, Lect. Notes Math., vol. 738, (1979), Springer-Verlag · Zbl 0417.57019 [2] Das, A. K.; Khare, S. S., Which Milnor manifolds bound, Indian J. Pure Appl. Math., 31, 11, 1503-1513, (2000) · Zbl 0967.57029 [3] Kostniowski, C.; Stong, R., \((\mathbb{Z}_2)^k\)-actions and characteristic numbers, Indiana Univ. Math. J., 28, 725-743, (1979) · Zbl 0437.57010 [4] Mukherjee, G.; Sankaran, P., Elementary abelian 2-group actions on flag manifolds and applications, Proc. Am. Math. Soc., 126, 2, 595-606, (1998) · Zbl 0889.57041 [5] Stong, R. E., Equivariant bordism and \((\mathbb{Z}_2)^k\)-actions, Duke Math. J., 37, 779-785, (1972) · Zbl 0204.23603 [6] tom Dieck, T., Fixpunkte vertauschbarer involutionen, Arch. Math., 20, 295-298, (1969) · Zbl 0207.53902 [7] tom Dieck, T., Characteristic numbers of G manifolds. I, Invent. Math., 13, 213-224, (1971) · Zbl 0216.45403 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.