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### The homology groups of small cover on a triangular prism and its number of characteristic functions

Huang Juhui 1,2

#### Affiliation(s)

1 Guilin Institute of Information Technology, Guilin, Guangxi, China
2 Baise University, Baise, Guangxi, China

Huang Juhui

#### ABSTRACT

Triangular prism is a common geometric shape. From the perspective of algebraic topology, it is a familiar simple convex polyhedron in algebraic topology. In this paper, we mainly calculate that there are only two kinds of characteristic functions on a triangular prism, and the homology groups of triangular prism is obtained by different characteristic functions are different. Firstly, according to the Morse function on the convex polytope Pn, We can give the cell decoposition of the corresponding small cover Mn over Pn, and the cellular chain complex {Di(Mn(λ)),∂i} of Mn. Secondly, considering the relationship between the boundary homomorphism {∂i} and the characteristic function λ, we can give the principle of how to determine the boundary homomorphism is given. Finally, the homology groups are computed by defination {Hi= ker∂i / Im∂i+1}, we can give the corresponding results.

#### KEYWORDS

Small cover, the homology group, triangular prism

#### CITE THIS PAPER

Huang Juhui, The homology groups of small cover on a triangular prism and its number of characteristic functions. Transactions on Computational and Applied Mathematics (2023) Vol. 3: 26-32. DOI: http://dx.doi.org/10.23977/tracam.2023.030104.

#### REFERENCES

[1] Davis M. W., Januskiewicz T. (1991) Convex Polytope, Coxeter Orbifolds and Torus Actions. Duke Mathematical Journal, 62, 417-451.
[2] Choi S., and Park H. (2017) On the Cohomology and Their Torsion of Real Toric Objects. Forum Mathematicum, 29, 543-553.
[3] Cai L., and Choi S, Y. (2016) On the Topology of Small Cover Associated to A Shellable Complex. Mathematics Algebraic Topology, 4, 1-30.
[4] Liu D. P. (2018) Cellular Chain Complex of Small Cover with Integer Coefficients and Its Application. Acta Mathematica Sinica, English Series, 34, 1742-1754.
[5] Choi S. (2008) The number of small covers over cubes. Algebraic and Geometric Topology, 8, 2391-2399.
[6] Liu Dengpin. The characteristic function of  and Moment-Angle manifold of Partial-quotient [D]: [PhD dissertation]. Shanghai: Department of Mathematics, Fudan University, 2012.
[7] Fu Xin. The small cover on the L bell polyhedron [D]: [Master's thesis]. Shanghai: Department of Mathematics, Fudan University, 2013.
[8] Hao Peide. The cohomology rigidity problem of the small cover on the L bell polyhedron and [D]: [Master's thesis]. Shanghai: Department of Mathematics, Fudan University, 2013.
[9] Lu Z., and Yu L. (2011) Topological Types of 3- Dimensional Small Covers. Forum Mathematicum, 23, 245-284.
[10] Lu Z., (2007) 2-Torus Manifolds Cobordism and Small Cover. Pracific Journal of Mathematics, 241, 285-308.