Weakly equivariant classification of small covers over a product of simplicies
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초록
Given a dimension function $\omega$, we introduce the notion of an $\omega$-vector weighted digraph and an $\omega$-equivalence between them. Then we establish a bijection between the weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of simplices $\Delta^{\omega(1)}\times\cdots \times \Delta^{\omega(m)}$ and the set of $\omega$-equivalence classes of $\omega$-vector weighted digraphs with $m$-labeled vertices, where $n$ is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of three simplices.
Abstract
Given a dimension function $\omega$, we introduce the notion of an $\omega$-vector weighted digraph and an $\omega$-equivalence between them. Then we establish a bijection between the weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of simplices $\Delta^{\omega(1)}\times\cdots \times \Delta^{\omega(m)}$ and the set of $\omega$-equivalence classes of $\omega$-vector weighted digraphs with $m$-labeled vertices, where $n$ is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of three simplices.
참고문헌 (10)
- M. Altunbulak, "On small covers over a product of simplices", Turkish J. Math, 42(3), pp.1528~1535 (2018)[DOI]
- M. Cai, "Small covers over prisms", Topology Appl, 154(11), pp.2228~2234 (2007)[DOI]
- S. Choi, "The number of small covers over cubes", Algebr. Geom. Topol, 8(4), pp.2391~2399 (2008)[DOI]
- S. Choi, "Classification of real Bott manifolds and acyclic digraphs", Trans. Amer. Math. Soc, 369(4), pp.2987~3011 (2017)[DOI]
- S. Choi, "Quasitoric manifolds over a product of simplices", Osaka J. Math, 47(1), pp.109~129 (2010)
- M. W. Davis, "Convex polytopes, Coxeter orbifolds and torus ac-tions", Duke Math. J, 62(2), pp.417~451 (1991)[DOI]
- D. G. Fon-Der-Flaass, "Local complementations of simple and directed graphs", Discrete Anal. Operations Research, 1, pp.15~34 (1996)
- A. Garrison, "Small covers of the dodecahedron and the 120-cell", Proc. Amer. Math. Soc, 131(3), pp.963~971 (2003)[DOI]
- A. Güçlükan Ilhan, "A note on small covers over cubes", Hacet. J. Math. Stat, 49(6), pp.1997~2006 (2020)[DOI]
- Z. Lü, "Equivariant classification of 2-torus manifolds", Colloq. Math, 115(2), pp.171~188 (2009)[DOI]