## 数学代写|拓扑学代写Topology代考|MATH315

2022年12月30日

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## 数学代写|拓扑学代写Topology代考|Cycle and Boundary Groups

Definition 2.25. (Cycle; Cycle group) A $p$-chain $c$ is a $p$-cycle if $\partial c=0$. In words, a chain that has empty boundary is a cycle. All p-cycles together form the $p$-th cycle group $Z_p$ under the addition that is used to define the chain groups. In terms of the boundary operator, $\mathbf{Z}p$ is the subgroup of $\mathbf{C}_p$ which is sent to the zero of $C{p-1}$, that is, $\operatorname{ker} \partial_p=Z_p$.

For example, in Figure 2.9(b), the 1-chain $a b+b c+c a$ is a 1-cycle since
$$\partial_1(a b+b c+c a)=(a+b)+(b+c)+(c+a)=0 .$$
Also, observe that the above 1-chain is the boundary of the triangle $a b c$. It is no accident that the boundary of a simplex is a cycle. Thanks to Proposition 2.8, the boundary of a $p$-chain is a $(p-1)$-cycle. This is a fundamental fact in homology theory.

The set of $(p-1)$-chains that can be obtained by applying the boundary operator $\partial_p$ on $p$-chains forms a subgroup of $(p-1)$-chains, called the ( $p-1)$-th boundary group $\mathrm{B}{p-1}=\partial_p\left(\mathrm{C}_p\right)$; or, in other words, the image of the boundary homomorphism is the boundary group, $\mathrm{B}{p-1}=\operatorname{im} \partial_p$. We have $\partial_{p-1} \mathrm{~B}{p-1}=0$ for $p>0$ due to Proposition $2.8$ and hence $\mathrm{B}{p-1} \subseteq \mathbf{Z}_{p-1}$. Figure $2.10$ illustrates cycles and boundaries.

## 数学代写|拓扑学代写Topology代考|Homology

The homology groups classify the cycles in a cycle group by putting together those cycles in the same class that differ by a boundary. From a group theoretic point of view, this is done by taking the quotient of the cycle groups with the boundary groups, which is allowed since the boundary group is a subgroup of the cycle group.

Definition 2.26. (Homology group) For $p \geq 0$, the $p$-th homology group is the quotient group $\mathrm{H}_p=\mathrm{Z}_p / \mathrm{B}_p$. Since we use a field, namely $\mathbb{Z}_2$, for coefficients, $\mathrm{H}_p$ is a vector space and its dimension is called the $p$-th Betti number, denoted by $\beta_p:$
$$\beta_p:=\operatorname{dim} \mathrm{H}_p .$$
Every element of $\mathrm{H}_p$ is obtained by adding a $p$-cycle $c \in Z_p$ to the entire boundary group, $c+\mathbf{B}_p$, which is a coset of $\mathbf{B}_p$ in $\mathbf{Z}_p$. All cycles constructed by adding an element of $\mathrm{B}_p$ to $c$ form the class $[c]$, referred to as the homology class of $c$. Two cycles $c$ and $c^{\prime}$ in the same homology class are called homologous, which also means $[c]=\left[c^{\prime}\right]$. By definition, $[c]=\left[c^{\prime}\right]$ if and only if $c \in c^{\prime}+\mathrm{B}_p$, and under $\mathbb{Z}_2$ coefficients, this also means that $c+c^{\prime} \in \mathbf{B}_p$. For example, in Figure $2.10$, the outer cycle $c_5$ is homologous to the sum $c_2+c_4$ because they together bound the 2 -chain consisting of all triangles. Also, obseerve that the group operation for $\mathrm{H}_p$ is defined by $[c]+\left[c^{\prime}\right]=\left[c+c^{\prime}\right.$.
Example 2.1. Consider the boundary complex $K$ of a tetrahedron which consists of four triangles, six edges, and four vertices. Consider the 0 -skeleton $\mathrm{K}^0$ of $K$ which consists of four vertices only (see Figure 2.11a). All four vertices whose classes coincide with them are necessary to generate $\mathrm{H}_0\left(K^0\right)$. Therefore, these four vertices form a basis of $\mathrm{H}_0\left(K^0\right)$. However, one can verify that $\mathrm{H}_0\left(K^1\right)$ for the 1-skeleton $K^1$ is generated by any one of the four vertices because all four vertices belong to the same class when we consider $K^1$. This exemplifies the fact that the rank of $\mathrm{H}_0(K)$ captures the number of connected components in a complex $K$.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Homology

$$\beta_p:=\operatorname{dim} \mathrm{H}_p .$$

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## MATLAB代写

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