### Action of boundary and co-boundary maps on a chain

__*Definition.*__ An abstract simplicial complex $K$ is a collection of finite sets that is closed
under set inclusion, i.e. if $\sigma \in K$ and $\tau \subseteq \sigma$, then $\tau \in K$.

__*Definition.*__ The boundary operator $\partial_d : C_d(K) \rightarrow C_{d-1}(K)$ is the linear function defined for each oriented $d$-simplex $\sigma = [v_0, ..., v_d]$ by
\begin{equation}
\partial_d (\sigma) = \partial_d [v_0, ..., v_d] = \sum_{i=0}^d (-1)^i [v_0, ..., \hat{v}_i, ..., v_d],
\end{equation}
where $[v_0, ... \hat{v}_i, ..., v_d]$ is the subset of $[v_0, ..., v_d]$ obtained by removing the vertex $v_i$.

Let $S_d(K)$ be the set of all oriented $d$-simplices of the simplicial complex $K$ (i.e. the set of basis elements of $C_d(K)$), and let $\tau in S_{d-1}(K)$. Then define the two sets
\begin{align}
S^+_d (K) &= \{ \sigma \in S_d(K) \, | \text{ the coefficient of } \tau \text{ in } \partial_d (\sigma) \text{ is } +1 \} \\
S^-_d (K) &= \{ \sigma \in S_d(K) \, | \text{ the coefficient of } \tau \text{ in } \partial_d (\sigma) \text{ is } -1 \}
\end{align}

__*Lemma (see [1])*__ Let $\partial^*_d$ be the adjoint of the boundary operator $\partial_d$, and let $\tau \in S_{d-1}(K)$. Then
\begin{equation}
\partial^*_d (\tau) = \sum_{\sigma' \in S^+_d (K, \tau)} \sigma' - \sum_{\sigma'' \in  S^-_d (K, \tau)} \sigma''.
\end{equation}


In this section we are going to see how boundary maps act on 


```python

```

## References <a class="anchor" id="refs"></a>

1. [R.Gustavson, Laplacians of Covering Complexes](https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1099&context=rhumj)
2. [From Topological Data Analysis to Deep Learning: No Pain No Gain](https://towardsdatascience.com/from-tda-to-dl-d06f234f51d)
3. [S. Mukherjee and J. Steenbergen, Random walks on simplicial complexes and harmonics](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5324709/)

4. [Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, Ali Jadbabaie, Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://arxiv.org/pdf/1807.05044.pdf)
5. [GUDHI Library](http://gudhi.gforge.inria.fr/doc/latest/index.html)

6. [R. Forman, Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature](https://link.springer.com/content/pdf/10.1007%2Fs00454-002-0743-x.pdf)

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examples/theory_simplicial_diffusion.ipynb
tsitsvero/hodgelaplacians
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2022-01-24T09:13:03.000Z
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tsitsvero/hodgelaplacians
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Qwen/Qwen-72B
1. YES
2. YES