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

: 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 P n , We can give the cell decoposition of the corresponding small cover M n over P n , and the cellular chain complex { D i ( M n( λ ) ), ∂ i } of M n . 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 { H i= ker ∂ i / Im ∂ i+1 }, we can give the corresponding results.


Foreword
The notion of small covers was firstly introduced by M. Davis and T. Jnuszkiewicz in [1] .An n-dimensional small cover is n-dimensional smooth manifold n M adimtting a n Z 2 -action and its orbit space is an n-dimensional simple convex polytope.We can construct a small cover    , n n P M over n P , and  is the characteristic function.Regarding the calculation of the homology of small cover, literature [2] mainly discusses the flexibility coefficient.Literature [3] was calculated using the cellular chain complex.Many further works has been carried out (see in [4][5][6][7][8][9][10] ).In this paper, the reference [3] cellular chain complex is used to calculate the homology group of small cover on the Triangular prism, and the corresponding results are obtained.

Prepare knowledge
Definition 2.1 [1] Setting n P is an n-dimensional convex polyhedron, and if each vertex of n P has n codimensions of 1, we call this convex polyhedron is single convex polyhedron.Definition 2.2 [1] Function n Z F  [1] (Morse function) The Morse function on a single convex polyhedron n P refers to a high flow function on a certain dimensional skeleton.Denoted

  v ind
as the number of flow directions at the vertex v on a single convex polyhedron n P .Setting .
Lemma 2.1 [1] n n P M  :  is a small cover on single convex polyhedra n P , setting be the single convex polyhedron n P on the small cover of the characteristic function  , we can write out on the cell chamber chain complex as: Lemma 2.2 [3] set is the small cover on a monoconvex polyhedron n P with characteristic function  , and the complex of the cellular chain is 


. We have the following conclusions: (1) In the following forms: ( An element above has the following form: 3. Homology group of small cover on the triangular prism

Combined structure of triangular prism and the cellular chain complex
The triangular prism is composed of five faces, and the five faces are On the prism satisfies any of the vertices of the triangular prism , And the determinant value of the characteristic function of any two edges at any vertex is 1  .There are only two characteristic functions of the five faces on the three prism: The second: Note: These two characteristic function are calculated in this article.Cellular chain complex on the triangular prism: Let be small cover of the characteristic function of the triangular prism, then the complex of the cellular chain complex is: The expression of the generating element is given as follows:  

Proof of the main conclusions
The first characteristic function of the five faces of the triangular prism: The homology group of small cover corresponding to the first characteristic function on the triangular prism is: is the standard base 2  e is calculated as follows: Because is the highest point of the flow direction of the vertex 4 e on the surface, it is taken 1 2 A as the base point, at this point we have: . , From the above calculation results know the characteristic function of each edge on 4 e , as shown in Figure 4 below:  Step 2: Calculate the edge homomorphism For odd numbers, For even numbers.
(2) the calculation of the 2  Order is an even number.
From the conclusions of Lemma 2.2: In summary, collated into the form of a matrix.

 
Step 3: Calculation of the homology group: Cellular chain complex on a triangular prism: Let be small cover of the characteristic function  of the triangular prism, then the complex of the cellular chain complex is: The homology group of the corresponding stained small cover   3  M on the triangular prism are: Following the above calculation method, the second characteristic function is used to calculate the homology group of small cover on the triangular prism: The second type of staining: According to the previous calculation method, the characteristic function of each side on the graph as shown in Figure 6: Corresponding corresponding stained  small cover on the triangular prism, the homology group is,

2 :
 is called characteristic function, if you meet the following conditions: any the vertices, have       1 ,....., , 2 1 is the height flow function (Morse function) on the triangular prism, calculating the index at each vertex:

Figure 1 :
Figure 1: Height flow function on three prism (Morse function)

4 e 4 e and surfaces that intersect with 4 e
group under type I characteristic function:Step 1: calculate the characteristic function of the edges, for example , the specific calculation process is as follows:Given the Morse flow of , as shown in Figure2below:

Figure 2
Figure 2: base of the edges, as shown in Figure3below:

Figure 3 :
Figure 3: Characteristic function of the surfaces 5 , 3 , 2 e e e Finally, the characteristic function of each upper edge on 4e is calculated as follows: Because Make a homomorphism as follows:

Figure 4 :
Figure 4: Characteristic function of each edge on the surface 4 e According to the previous calculation method, each side on the surface 5e is marked on the graph, as shown in Figure5below:

Figure 5 :
Figure 5: Characteristic function of each edge on the surface 5 e

Figure 6 :
Figure 6: Characteristic function of each edge on the surface value of the characteristic function of any two edges that intersect at any vertex is positive 1 or negative 1.There are only two characteristic function of the five faces on the triangular prism, and the results of the homology group of small cover on the triangular prism are not consistent.The first:    