# 数学代写|表示论代写Representation theory代考|MAST90017

#### Doug I. Jones

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## 数学代写|表示论代写Representation theory代考|Elementary Inclusion

Consider an elementary inclusion $K_i \hookrightarrow K_{i+1}$. Assume that $K_i$ has a valid annotation. We describe how we obtain a valid annotation for $K_{i+1}$ from that of $K_i$ after inserting the $p$-simplex $\sigma=K_{i+1} \backslash K_i$. We compute the annotation $\mathrm{a}_{\partial \sigma}$ for the boundary $\partial \sigma$ in $K_i$ and take actions as follows which ultimately lead to computing the persistence diagram.

Case (i): If $\mathrm{a}{\partial \sigma}$ is a zero vector, the class $[\partial \sigma]$ is trivial in $\mathrm{H}{p-1}\left(K_i\right)$. This means that $\sigma$ creates a $p$-cycle in $K_{i+1}$ and by duality a $p$-cocycle is killed while going left from $K_{i+1}$ to $K_i$. In this case we augment the annotations for all $p$-simplices by one element with a timestamp $i+1$, that is, the annotation $\left[b_1, b_2, \ldots, b_g\right.$ ] for every $p$-simplex $\tau$ is updated to $\left[b_1, b_2, \ldots, b_g, b_{g+1}\right]$ with $b_{g+1}$ being timestamped $i+1$. We set $b_{g+1}=0$ for $\tau \neq \sigma$ and $b_{g+1}=1$ for $\tau=\sigma$. The element $b_i$ of $\mathrm{a}_\sigma$ is set to zero for $1 \leq i \leq g$. Other annotations for other simplices remain unchanged. See Figure 4.2(a).

Case (ii): If $\mathrm{a}{\partial \sigma}$ is not a zero vector, the class of the $(p-1)$-cycle $\partial \sigma$ is nontrivial in $\mathrm{H}{p-1}\left(K_i\right)$. Therefore, $\sigma$ kills the class of this $(p-1)$-cycle and a corresponding class of $(p-1)$-cocycles is born in the reverse direction. We simulate it by forcing $\mathrm{a}{\partial \sigma}$ to be zero which affects other annotations as well. Let $i_1{i_1}, b_{i_2}, \ldots, b_{i_k}$ are all of the nonzero elements in $\mathrm{a}{\partial \sigma}=$ $\left[b_1, b_2, \ldots, b{i_k}, \ldots, b_g\right]$. Recall that $\phi_j$ denotes the $(p-1)$-cocycle given by its evaluation $\phi_j\left(\sigma^{\prime}\right)=\mathrm{a}{\sigma^{\prime}}[j]$ for every $(p-1)$-simplex $\sigma^{\prime} \in K_i$ (Proposition 4.5). With this notation, the cocycle $\phi=\phi{i_1}+\phi_{i_2}+\cdots+\phi_{i_k}$ is born after deleting $\sigma$ in the reverse direction. This cocycle does not exist after time $i_k$ in the reverse direction. In other words, the cohomology class $[\phi]$ which is born leaving the time $i+1$ is killed at time $i_k$. This pairing matches that of the standard persistence algorithm where the youngest basis element is chosen to be paired among all those whose combination is killed. We add the vector $\mathrm{a}{\partial \sigma}$ to the annotation of every $(p-1)$-simplex whose $i_k$-th element is nonzero. This zeroes out the $i_k$-th element of the annotation for every ( $p-1$ )-simplex and at the same time updates other elements so that a valid annotation according to Proposition $4.5$ is maintained. We simply delete the $i_k$-th element from the annotation for every ( $p-1)$-simplex. See Figure 4.2(b). We further set the annotation $\mathrm{a}\sigma$ for $\sigma$ to be a zero vector of length $s$, where $s$ is the length of the annotation vector of every $p$-simplex at this point.

## 数学代写|表示论代写Representation theory代考|Elementary Collapse

The case for handling collapse is more interesting. It has three distinct steps: (i) elementary inclusions to satisfy the so-called link condition; (ii) local annotation transfer to prepare for the collapse; and (iii) collapse of the simplices with updated annotations. We explain each of these steps now.

The elementary inclusions that may precede the final collapse are motivated by a result that connects collapses with the change in cohomology. Consider an elementary collapse $K_i \stackrel{f_i}{\rightarrow} K_{i+1}$ where the vertex pair $(u, v)$ collapses to $u$. The following link condition, introduced in [121] and later used to preserve homotopy [12], becomes relevant.

Definition 4.8. (Link condition) A vertex pair $(u, v)$ in a simplicial complex $K_i$ satisfies the link condition if the edge $u v \in K_i$ and $\mathrm{Lk} u \cap \mathrm{Lk} v=\mathrm{Lk} u v$. An elementary collapse $f_i: K_i \rightarrow K_{i+1}$ satisfies the link condition if the vertex pair on which $f_i$ is not injective satisfies the link condition.

Proposition 4.7. [12] If an elementary collapse $f_i: K_i \rightarrow K_{i+1}$ satisfies the link condition, then the underlying spaces $\left|K_i\right|$ and $\left|K_{i+1}\right|$ remain homotopy equivalent. Hence, the induced homomorphisms $f_{i }: \mathrm{H}p\left(K_i\right) \rightarrow \mathrm{H}_p\left(K{i+1}\right)$ and $f_i^: \mathrm{H}^p\left(K_i\right) \leftarrow \mathrm{H}^p\left(K_{i+1}\right)$ are isomorphisms.

If an elementary collapse satisfies the link condition, we can perform the collapse knowing that the cohomology does not change. Otherwise, we know that the cohomology is affected by the collapse and it should be reflected in our updates for annotations. The diagram below provides a precise means to carry out the change in cohomology. Let $S$ be the minimal set of simplices ordered in nondecreasing order of their dimensions whose addition to $K_i$ makes $(u, v)$ satisfy the link condition. Une can describe a construction of $S$ recursively as follows. In dimension one, if the edge $(u, v)$ is missing, it is added to $S$.

# 表示论代考

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