## 数学代写|拓扑学代写Topology代考|MATH3531

2022年12月30日

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## 数学代写|拓扑学代写Topology代考|Chains

Let $K$ be a simplicial $k$-complex with $m_p$ number of $p$-simplices, $k \leq p \leq 0$. A $p$-chain $c$ in $K$ is a formal sum of $p$-simplices added with some coefficients, that is, $c=\sum_{i=1}^{m_p} \alpha_i \sigma_i$ where $\sigma_i$ are the $p$-simplices and $\alpha_i$ are the coefficients. Two $p$-chains $c=\sum \alpha_i \sigma_i$ and $c^{\prime}=\sum \alpha_i^{\prime} \sigma_i$ can be added to obtain another $p$-chain:
$$c+c^{\prime}=\sum_{i=1}^{m_p}\left(\alpha_i+\alpha_i^{\prime}\right) \sigma_i$$
In general, coefficients can come from a ring $R$ with its associated additions making the chains constituting an $R$-module. For example, these additions can be integer additions where the coefficients are integers; for example, from two 1-chains (edges) we get
$$\left(2 e_1+3 e_2+5 e_3\right)+\left(e_1+7 e_2+6 e_4\right)=3 e_1+10 e_2+5 e_3+6 e_4 .$$
Notice that while writing a chain, we only write the simplices that have nonzero coefficient in the chain. We follow this convention all along. In our case, we will focus on the cases where the coefficients come from a field $\mathbf{k}$. In particular, we will mostly be interested in $\mathbf{k}=\mathbb{Z}$ 2. This means that the coefficients come from the field $\mathbb{Z}_2$ whose elements can only he 0 or 1 with the modnlo- 2 additions $0+0=0,0+1=1$, and $1+1=0$. This gives us $\mathbb{Z}_2$-additions of chains; for example, we have
$$\left(e_1+e_3+e_4\right)+\left(e_1+e_2+e_3\right)=e_2+e_4 .$$
Observe that $p$-chains with $\mathbb{Z}_2$-coefficients can be treated as sets: the chain $e_1+e_3+e_4$ is the set $\left{e_1, e_3, e_4\right}$, and $\mathbb{Z}_2$-addition between two chains is simply the symmetric difference between the corresponding sets.

From now on, unless specified otherwise, we will consider all chain additions to be $\mathbb{Z}_2$-additions. One should keep in mind that one can have parallel concepts for coefficients and additions coming from integers, reals, rationals, fields, and other rings. Under $\mathbb{Z}2$-additions, we have $$c+c=\sum{i=1}^{m_p} 0 \sigma_i=0 .$$

## 数学代写|拓扑学代写Topology代考|Boundaries and Cycles

The chain groups at different dimensions are related by a boundary operator. Given a $p$-simplex $\sigma=\left{v_0, \ldots, v_p\right}$ (also denoted as $v_0 v_1 \cdots v_p$ ), let
$$\partial_p \sigma=\sum_{i=0}^p\left{v_0, \ldots, \hat{v}_i, \ldots, v_p\right}$$
where $\hat{v_i}$ indicates that the vertex $v_i$ is omitted. Informally, we can view $\partial_p$ as a map that sends a $p$-simplex $\sigma$ to the $(p-1)$-chain that has nonzero coefficients only on $\sigma$ ‘s $(p-1)$-faces, also referred to as $\sigma$ ‘s boundary. At this point, it is instructive to note that the boundary of a vertex is empty, that is, $\partial_0 \sigma=\varnothing$.

Extending $\partial_p$ to a $p$-chain, we obtain a homomorphism $\partial_p: \mathbf{C}p \rightarrow \mathbf{C}{p-1}$ called the boundary operator that produces a $(p-1)$-chain when applied to a p-chain:
$$\partial_p c=\sum_{i=1}^{m_p} \alpha_i\left(\partial_p \sigma_i\right) \text { for a } p \text {-chain } c=\sum_{i=1}^{m_p} \alpha_i \sigma_i \in \mathbf{C}p .$$ Again, we note the special case of $p=0$ when we get $\partial_0 c=\varnothing$. The chain group $C{-1}$ has only one single element which is its identity 0 . On the other hand, we also assume that if $K$ is a $k$-complex, then $\mathrm{C}_p$ is 0 for $p>k$.
Consider the complex in Figure 2.9(b). For the 2-chain $a b c+b c d$ we get
$$\partial_2(a b c+b c d)=(a b+b c+c a)+(b c+c d+d b)=a b+c a+c d+d b .$$
It means that from the two triangles sharing the edge $b c$, the boundary operator returns the four boundary edges that are not shared. Similarly. one can check that the boundary of the 2-chains consisting of all three triangles in Figure 2.9(b) contains all seven edges. In particular, the edge $b c$ does not get cancelled because all three (odd) triangles adjoin it:
$$\partial_2(a b c+b c d+b c e)=a b+b c+c a+b e+c e+b d+d c .$$
One important property of the boundary operator is that applying it twice produces an empty chain.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Chains

$$c+c^{\prime}=\sum_{i=1}^{m_p}\left(\alpha_i+\alpha_i^{\prime}\right) \sigma_i$$

$$\left(2 e_1+3 e_2+5 e_3\right)+\left(e_1+7 e_2+6 e_4\right)=3 e_1+10 e_2$$

$$\left(e_1+e_3+e_4\right)+\left(e_1+e_2+e_3\right)=e_2+e_4$$

$e_1+e_3+e_4$ 是集合 Meft{e_1, e_3, e_4lright $}$ ，和 $\mathbb{Z}_2-$

$$c+c=\sum i=1^{m_p} 0 \sigma_i=0 .$$

## 数学代写|拓扑学代写Topology代考|Boundaries and Cycles

$$\partial_2(a b c+b c d)=(a b+b c+c a)+(b c+c d+d b)$$

$$\partial_2(a b c+b c d+b c e)=a b+b c+c a+b e+c e+b d$$

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