Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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WEAKLY EQUIVARIANT CLASSIFICATION OF SMALL COVERS OVER A PRODUCT OF SIMPLICIES

  • Received : 2022.03.04
  • Accepted : 2022.07.04
  • Published : 2022.09.01
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Abstract

Given a dimension function 𝜔, we introduce the notion of an 𝜔-vector weighted digraph and an 𝜔-equivalence between them. Then we establish a bijection between the weakly (ℤ/2)n-equivariant homeomorphism classes of small covers over a product of simplices ∆𝜔(1) × ⋯ × ∆𝜔(m) and the set of 𝜔-equivalence classes of 𝜔-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 (ℤ/2)n-equivariant homeomorphism classes of small covers over a product of three simplices.

Keywords

Acknowledgement

This work is supported by The Scientific and Technological Research Council of Turkey (Grant No: TBAG/118F506).

References

  1. M. Altunbulak and A. Guclukan Ilhan, On small covers over a product of simplices, Turkish J. Math. 42 (2018), no. 3, 1528-1535. https://doi.org/10.3906/mat-1706-40
  2. M. Cai, X. Chen, and Z. Lu, Small covers over prisms, Topology Appl. 154 (2007), no. 11, 2228-2234. https://doi.org/10.1016/j.topol.2007.02.008
  3. S. Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008), no. 4, 2391-2399. https://doi.org/10.2140/agt.2008.8.2391
  4. S. Choi, M. Masuda, and S. Oum, Classification of real Bott manifolds and acyclic digraphs, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2987-3011. https://doi.org/10.1090/tran/6896
  5. S. Choi, M. Masuda, and D. Y. Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. 47 (2010), no. 1, 109-129. http://projecteuclid.org/euclid.ojm/1266586788
  6. M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417-451. https://doi.org/10.1215/S0012-7094-91-06217-4
  7. D. G. Fon-Der-Flaass, Local complementations of simple and directed graphs, Discrete Anal. Operations Research 1 (1996), 15-34. https://doi.org/10.1007/978-94-009-1606-7_3
  8. A. Garrison and R. Scott, Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc. 131 (2003), no. 3, 963-971. https://doi.org/10.1090/S0002-9939-02-06577-2
  9. A. Guclukan Ilhan, A note on small covers over cubes, Hacet. J. Math. Stat. 49 (2020), no. 6, 1997-2006. https://doi.org/10.15672/hujms.631676
  10. Z. Lu and M. Masuda, Equivariant classification of 2-torus manifolds, Colloq. Math. 115 (2009), no. 2, 171-188. https://doi.org/10.4064/cm115-2-3