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**The number of small covers over cubes and the product of at most three simplices up to equivariant cobordism.**
*(English)*
Zbl 1231.57028

This paper considers the number of equivariant cobordism classes of small covers over the product of simplices. We recall that a small cover, defined by M. W. Davis and T. Januszkiewicz in [Duke Math. J. 62, No.2, 417–451 (1991; Zbl 0733.52006)], is a smooth closed manifold \(M^n\) with a locally standard \((\mathbb{Z}_2)^n\)-action such that its orbit space is a simple convex polytope.

Denoting by \(\mathcal{M}_n\) the set of equivariant unoriented cobordism classes of all \(n\)-dimensional small covers, let \(\mathcal{M}_* = \sum_{n\geq 1}\mathcal{M}_n\), which is generated by the classes of small covers over the product of simplices. Since the equivariant cobordism class of a small cover over a simple convex polytope is determined by its tangential representation set, which can be identified with the characteristic function of the simple convex polytope, the authors using this function determine relevant results in what concerns the number of small covers over cubes and the product of at most three simplices up to equivariant cobordism.

Denoting by \(\mathcal{M}_n\) the set of equivariant unoriented cobordism classes of all \(n\)-dimensional small covers, let \(\mathcal{M}_* = \sum_{n\geq 1}\mathcal{M}_n\), which is generated by the classes of small covers over the product of simplices. Since the equivariant cobordism class of a small cover over a simple convex polytope is determined by its tangential representation set, which can be identified with the characteristic function of the simple convex polytope, the authors using this function determine relevant results in what concerns the number of small covers over cubes and the product of at most three simplices up to equivariant cobordism.

Reviewer: Alice Kimie Miwa Libardi (Sao Paulo)

### Citations:

Zbl 0733.52006
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\textit{Y. Chen} and \textit{Y. Wang}, Proc. Japan Acad., Ser. A 87, No. 6, 95--98 (2011; Zbl 1231.57028)

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### References:

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[4] | M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417-451. · Zbl 0733.52006 |

[5] | Z. Lü, 2-torus manifolds, cobordism and small covers, Pacific J. Math. 241 (2009), no. 2, 285-308. · Zbl 1181.57036 |

[6] | Z. Lü and Q. Tan, A differential operator and tom Dieck-Kosniowski-Stong localization theorem, arXiv: |

[7] | R. E. Stong, Equivariant bordism and \((Z_{2})^{k}\) actions, Duke Math. J. 37 (1970), 779-785. · Zbl 0204.23603 |

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