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# 数学代写|拓扑学代写TOPOLOGY代考|The Boundary Map

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## 数学代写|拓扑学代写TOPOLOGY代考|The Boundary Map

Recall that the idea of homology is to count holes that are not boundaries. Thus, we need to be able to detect when something is or is not a boundary. We start, naturally enough, by defining the boundary of a simplex.

Suppose that we have an $n$-simplex $\left[v_0, \ldots, v_n\right]$. We define its boundary to be
$$\partial_n\left(\left[v_0, \ldots, v_n\right]\right)=\sum_{i=0}^n(-1)^i\left[v_0, \ldots, \widehat{v}i, \ldots, v_n\right]$$ where the hatted term $\widehat{v}_i$ is omitted. For example, if $n=2$, we have $$\partial_2\left(\left[v_0, v_1, v_2\right]\right)=\left[v_1, v_2\right]-\left[v_0, v_2\right]+\left[v_0, v_1\right]$$ Observe that the boundary of an $n$-simplex is an $(n-1)$-chain. We now extend the boundary map to a homomorphism $\partial_n: C_n(X) \rightarrow C{n-1}(X)$ in the only way possible:
$$\partial_n\left(a_1 T_1+\cdots+a_r T_r\right)=a_1 \partial_n\left(T_1\right)+\cdots+a_r \partial_n\left(T_r\right)$$
The most important property of the boundary map is that the boundary of a boundary is zero, i.e. the following theorem.

Theorem 13.2 If $A \in C_n(X)$ is any $n$-chain, then $\partial_{n-1} \circ \partial_n(A)=0$.
Proof We simply compute. It suffices to check this in the case that $A=\left[v_0, \ldots, v_n\right]$ is an $n$-simplex because the boundary maps are homomorphisms. Then we have
\begin{aligned} \partial_{n-1} \circ \partial_n(A)= & \partial_{n-1}\left(\sum_{i=0}^n(-1)^i\left[v_0, \ldots, \widehat{v}i, \ldots, v_n\right]\right) \ = & \sum{i=0}^n(-1)^i \partial_{n-1}\left(\left[v_0, \ldots, \widehat{v}i, \ldots, v_n\right]\right) \ = & \sum{i=0}^n(-1)^i\left(\sum_{j=0}^{i-1}(-1)^j\left[v_0, \ldots, \widehat{v}j, \ldots, \widehat{v}_i, \ldots, v_n\right]\right. \ & \left.+\sum{j=i+1}^n(-1)^{j-1}\left[v_0, \ldots, \widehat{v}_i, \ldots, \widehat{v}_j, \ldots, v_n\right]\right) . \end{aligned}

In the final expression on the right, we see that there are exactly two terms missing both $v_i$ and $v_j$. But what are the signs? Suppose that $i<j$. Then it occurs once with coefficient $(-1)^i(-1)^{j-1}=(-1)^{i+j-1}$ by removing first $v_i$ and then $v_j$, and once with coefficient $(-1)^j(-1)^i=(-1)^{i+j}$ by first removing $v_j$ and then removing $v_i$. The sum of these two coefficients is 0 , so the coefficient of the $(n-2)$-simplex $\left[v_0, \ldots, \widehat{v}i, \ldots, \widehat{v}_j, \ldots, v_n\right]$ is 0 . This is true for each $(n-2)$-simplex, so $\partial{n-1} \circ$ $\partial_n(A)=0$, as desired.

## 数学代写|拓扑学代写TOPOLOGY代考|Homology

At last, we can define homology. We know that $B_n(X) \leq Z_n(X)$; the homology is a measure of the discrepancy between these two groups.

Definition 13.6 The $n^{\text {th }}$ homology group of $X$ is the quotient group $H_n(X)=\frac{Z_n(X)}{B_n(X)}$.
It is true, although we will not prove it here, that the homology groups do not depend on the choice of triangulation of $X$. (See [Mun84, Section 18] for a proof.) Hence, the homology groups are a homeomorphism invariant. In fact, they are also a homotopy invariant: two spaces that are homotopy equivalent have the same homology groups. Furthermore, the triangulation does not have to satisfy our strict rules for triangulations as defined in Chapter 3. Instead, simplices are allowed to meet in more complicated ways, without any ill effects. In particular, the additional possibilities we allow are that lower-dimensional faces of a simplex can be glued together, and the intersection of two simplices must be a union of their lower-dimensional faces. In addition, we are allowed to glue parts of the boundary of a simplex together, as long as they are of the same type: that is, we are allowed to glue together two vertices of a simplex, or two edges, and so forth. The official name of a triangulation with these weaker rules about intersection types is a $\Delta$-complex, and perhaps we should correspondingly call our homology $\Delta$-homology. But this name is less commonly used than simplicial homology, so we will stick to the term simplicial homology.
A very simple example of a $\Delta$-complex that takes advantage of our looser rules for intersection types is a (filled-in) triangle with two vertices glued together, as shown in Figure 13.2. This isn’t a simplicial complex or a triangulation in the usual sense, but we allow it among our $\Delta$-complexes and could compute its homology if desired.

## 数学代写|拓扑学代写TOPOLOGY代考|The Boundary Map

$$\partial_n\left(\left[v_0, \ldots, v_n\right]\right)=\sum_{i=0}^n(-1)^i\left[v_0, \ldots, \hat{v} i, \ldots, v_n\right]$$

$$\partial_{n-1} \circ \partial_n(A)=\partial_{n-1}\left(\sum_{i=0}^n(-1)^i\left[v_0, \ldots, \hat{v} i, \ldots, v_n\right]\right)=\sum i=0^n(-1)^i \partial_{n-1}\left(\left[v_0, \ldots, \hat{v} i, \ldots, v_n\right]\right)=\sum i=0^n(-1)^i$$

## MATLAB代写

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