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## 数学代写|拓扑学代写Topology代考|Induced Homology

Continuous functions from a topological space to another topological space take cycles to cycles and boundaries to boundaries. Therefore, they induce a map in their homology groups as well. Here we will restrict ourselves only to simplicial complexes and simplicial maps that are the counterpart of continuous maps between topological spaces. Simplicial maps between simplicial complexes take cycles to cycles and boundaries to boundaries with the following definition.

Definition 2.27. (Chain map) Let $f: K_{1} \rightarrow K_{2}$ be a simplicial map. The chain map $f_{#}: \mathrm{C}{p}\left(K{1}\right) \rightarrow \mathrm{C}{p}\left(K{2}\right)$ corresponding to $f$ is defined as follows. If $c=\sum \alpha_{i} \sigma_{i}$ is a $p$-chain, then $f_{#}(c)=\sum \alpha_{i} \tau_{i}$ where
$$\tau_{i}= \begin{cases}f\left(\sigma_{i}\right), & \text { if } f\left(\sigma_{i}\right) \text { is a } p \text {-simplex in } K_{2}, \ 0, & \text { otherwise. }\end{cases}$$
For example, in Figure $2.12$, the 1 -cycle $b c+c d+d b$ in $K_{1}$ is mapped to the 1 -chain $e g+e g=0$ by the chain map $f_{#}$.

Proposition 2.9. Let $f: K_{1} \rightarrow K_{2}$ be a simplicial map. Let $\partial_{p}^{K_{1}}$ and $\partial_{p}^{K_{2}}$ denote the boundary homomorphisms in dimension $p \geq 0$. Then, the induced chain maps commute with the boundary homomorphisms, that is, $f_{#} \circ \partial_{p}^{K_{1}}=$ $\partial_{p}^{K_{2}} \circ f_{#}$

The statement in the above proposition can also be represented with the following diagram, which we say commutes since starting from the top left corner, one reaches the same chain at the lower right corner using both paths first going right and then down, or first going down and then right (see Definition $3.15$ in the next chapter).

## 数学代写|拓扑学代写Topology代考|Singular Homology

So far we have considered only simplicial homology which is defined on a simplicial complex without any assumption of a particular topology. Now, we extend this definition to topological spaces. Let $X$ be a topological space. We bring the notion of simplices in the context of $X$ by considering maps from the standard $d$-simplices to $X$. A standard $p$-simplex $\Delta^{p}$ is defined by the convex hull of $p+1$ points $\left{\left(x_{1}, \ldots, x_{i}, \ldots, x_{p+1}\right) \mid x_{i}=1\right.$ and $x_{j}=0$ for $j \neq i}_{i=1, \ldots, p+1}$ in $\mathbb{R}^{p+1}$.

Definition 2.28. (Singular simplex) A singular $p$-simplex for a topological space $X$ is defined as a map $\sigma: \Delta^{p} \rightarrow X$.

Notice that the map $\sigma$ need not be injective and thus $\Delta^{p}$ may be “squashed” arbitrarily in its image. Nevertheless, we can still have a notion of the chains, boundaries, and cycles which are the main ingredients for defining a homology group called the singular homology of $X$.

The boundary of a $p$-simplex $\sigma$ is given by $\partial \sigma=\tau_{0}+\tau_{2}+\cdots+\tau_{p}$ where $\tau_{i}:\left(\partial \Delta^{p}\right){i} \rightarrow X$ is the restriction of the map $\sigma$ on the $i$-th facet $\left(\partial \Delta^{p}\right){i}$ of $\Delta^{p}$

A $p$-chain is a sum of singular $p$-simplices with coefficients from integers, reals, or some appropriate rings. As before, under our assumption of $\mathbb{Z}{2}$ coefficients, a singular $p$-chain is given by $\sum{i} \alpha_{i} \sigma_{i}$ where $\alpha_{i}=0$ or 1 . The boundary of a singular $p$-chain is defined the same way as we did for simplicial chains, the only difference being that we have to accommodate for infinite chains:
$$\partial\left(c_{p}=\sigma_{1}+\sigma_{2}+\cdots\right)=\partial \sigma_{1}+\partial \sigma_{2}+\cdots$$
We get the usual chain complex with $\partial_{p} \circ \partial_{p-1}=0$ for all $p>0$
$$\ldots \stackrel{\partial_{p+1}}{\rightarrow} \mathrm{C}{p} \stackrel{\partial{p}}{\rightarrow} \mathrm{C}{p-1} \stackrel{\partial{p-1}}{\rightarrow} \cdots$$
and can define the cycle and boundary groups as $Z_{p}=$ ker $\partial_{p}$ and $\mathbf{B}{p}=$ $\operatorname{im} \partial{p+1}$. We have the singular homology defined as the quotient group $\mathrm{H}{p}=$ $Z{p} / \mathrm{B}_{p}$

A useful fact is that singular and simplicial homology coincide when both are well defined.

## 数学代写|拓扑学代写Topology代考|Cochains, Coboundaries, and Cocycles

A $p$-cochain is a homomorphism $\phi: \mathrm{C}{p} \rightarrow \mathbb{Z}{2}$ from the chain group to the coefficient ring over which $C_{p}$ is defined which is $\mathbb{Z}{2}$ here. In this case, a $p$ cochain $\phi$ is given by its evaluation $\phi(\sigma)(0$ or 1$)$ on every $p$-simplex $\sigma$ in $K$, or more precisely, a $p$-chain $c=\sum{i=1}^{m_{p}} \alpha_{i} \sigma_{i}$ gets a value
$$\phi(c)=\alpha_{1} \phi\left(\sigma_{1}\right)+\alpha_{2} \phi\left(\sigma_{2}\right)+\cdots+\alpha_{m_{p}} \phi\left(\sigma_{m_{p}}\right)$$
Also, verify that $\phi\left(c+c^{\prime}\right)=\phi(c)+\phi\left(c^{\prime}\right)$ satisfying the property of group homomorphism. For a chain $c$, the particular cochain that assigns 1 to a simplex if and only if it has a nonzero coefficient in $c$ is called its dual cochain $c^{*}$. The $p$-cochains form a cochain group $C^{p}$ dual to $C_{p}$ where the addition is defined by $\left(\phi+\phi^{\prime}\right)(c)=\phi(c)+\phi^{\prime}(c)$ by taking $\mathbb{Z}{2}$-addition on the right. We can also define a scalar multiplication $(\alpha \phi)(c)=\alpha \phi(c)$ by using the $\mathbb{Z}{2}$-multiplication. This makes $C^{p}$ a vector space.

Similar to boundaries of chains, we have the notion of coboundaries of cochains $\delta_{p}: \mathrm{C}^{p} \rightarrow \mathrm{C}^{p+1}$. Specifically, for a $p$-cochain $\phi$, its $(p+1)$ coboundary is given by the homomorphism $\delta \phi: \mathrm{C}^{p+1} \rightarrow \mathbb{Z}{2}$ defined as $\delta \phi(c)=\phi(\partial c)$ for any $(p+1)$-chain $c$. Therefore, the coboundary operator $\delta$ takes a $p$-cochain and produces a $(p+1)$-cochain giving the sequence for a simplicial $k$-complex: $$0=\mathrm{C}^{-1} \stackrel{\delta{-1}}{\longrightarrow} \mathrm{C}^{0} \stackrel{\delta_{0}}{\rightarrow} \mathrm{C}^{1} \stackrel{\delta_{1}}{\rightarrow} \cdots \stackrel{\delta_{k-1}}{\longrightarrow} \mathrm{C}^{k} \stackrel{\delta_{k}}{\rightarrow} \mathrm{C}^{k+1}=0 .$$

## 数学代写|拓扑学代写Topology代考|Induced Homology

$$\tau_{i}=\left{f\left(\sigma_{i}\right), \quad \text { if } f\left(\sigma_{i}\right) \text { is a } p \text {-simplex in } K_{2}, 0, \quad\right. \text { otherwise. }$$

## 数学代写|拓扑学代写Topology代考|Singular Homology

\eft 的分隔符缺失或无法识别 在 $\mathbb{R}^{p+1}$.

$$\partial\left(c_{p}=\sigma_{1}+\sigma_{2}+\cdots\right)=\partial \sigma_{1}+\partial \sigma_{2}+\cdots$$

$$\ldots \stackrel{\partial_{p+1}}{\rightarrow} \mathrm{C} p \stackrel{\partial p}{\rightarrow} \mathrm{C} p-1 \stackrel{\partial p-1}{\rightarrow} \cdots$$

## 数学代写|拓扑学代写Topology代考|Cochains, Coboundaries, and Cocycles

$$\phi(c)=\alpha_{1} \phi\left(\sigma_{1}\right)+\alpha_{2} \phi\left(\sigma_{2}\right)+\cdots+\alpha_{m_{p}} \phi\left(\sigma_{m_{p}}\right)$$

$$0=\mathrm{C}^{-1} \stackrel{\delta-1}{\longrightarrow} \mathrm{C}^{0} \stackrel{\delta_{0}}{\rightarrow} \mathrm{C}^{1} \stackrel{\delta_{1}}{\rightarrow} \cdots \stackrel{\delta_{k-1}}{\longrightarrow} \mathrm{C}^{k} \stackrel{\delta_{k}}{\rightarrow} \mathrm{C}^{k+1}=0$$

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