How does the high-dimensional combinatorial Laplacian work?

Question

When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,v_{i-1},v_{i+1},...,v_k],$$ and the coboundary map $\delta_k$ based on the space of cochains $C^k(X)$ is the adjoint operator of $\partial_{k+1}$, i.e. $$(\delta_k f)([v_0,...,v_{k+1}])=f(\partial_{k+1}[v_0,...,v_{k+1}]) =\sum_{i=0}^k (-1)^if([v_0,...,v_{i-1},v_{i+1},...,v_{k+1}]).$$

See https://magnus.ece.gatech.edu/Papers/MuhammadEgerstedtMTNS06.pdf page 4.

In the paper, the authors define the $k$-combinatorial Laplacians $\Delta_k:C^k(X;\mathbb{R})\to C^k(X;\mathbb{R})$ and $\mathcal{L}_k:C_k(X;\mathbb{R})\to C_k(X;\mathbb{R})$ by $$\Delta_k=\delta_{k-1} \delta^*_{k-1}+\delta^*_k\delta_k $$ $$\mathcal{L}_k=\partial_{k+1} \partial^*_{k+1}+\partial^*_k\partial_k $$

Here is my question:

For me, neither of these two $k$-combinatiorial Laplacians seem well-defined. For example, $\delta_k$ works on a cochain $f\in C^k(X;\mathbb{R})$ and after that we get a cochain in $C^{k+1}(X;\mathbb{R})$, but how can we proceed with $\delta^*_k$? Since $\delta^*_k = \partial_{k+1}$, how can we put $\partial_{k+1}$ on a cochain?

Can you give me an example of the $\delta^*_k\delta_k$ on any cochain $f$?

For instance, what is, $\delta^*_1\delta_1 f([v_0,v_1])= \partial_2\delta_1 f([v_0,v_1])$?

Answer 1

A lot is lost in the abstract definitions of coboundary maps and cohomology (at least in the finite dimensional case, which I'm restricting to in my answer). But of course, any finite dimensional vector space with basis $e_1, \dots e_n$ is isomorphic to its dual via the map sending $e_i$ to the map that is $1$ on $e_i$, $0$ on other basis vectors. See the Wikipedia article.

Now the coboundary map is the transpose of the boundary map. Working through, you get in a simplicial complex that for a face $\sigma$ we have $\delta(\sigma) = \sum (-1)^* \sigma\cup\{v\}$. Here, the sum is over all vertices $v$ so that $\sigma \cup \{v\}$ is a face, and the sign depends on which position $v$ appears when $\sigma \cup \{v\}$ is sorted by index. To say it another way: the boundary map sends $\sigma$ to a linear combination of all vertex-deletions of $\sigma$, with appropriate signs. The coboundary map sends $\sigma$ to all possible vertex-additions of $\sigma$, with appropriate signs.

An alternative model of cohomology of a simplicial complex that I like quite a bit is via the "exterior Stanley-Reisner ring", and works as follows. Given a simplicial complex $\Delta$ on vertices $e_0,e_1,\dots,e_n$, begin with the exterior algebra $\Lambda[e_0,e_1,\dots,e_n]$ generated over variables corresponding to the vertices. Now mod out by the ideal $I$ generated by all non-faces of $\Delta$. That is, if $\{w_1,\dots,w_k\}$ is a set of vertices that is not a face of $\Delta$, then then add $w_1 \wedge \cdots \wedge w_k$ to the set of generators of $I$. (This works since simplicial complexes are closed under inclusion, while ideals are closed under multiplications.) You can easily see that the $k$-graded part is isomorphic to $C_k$, hence to $C^k$. Now the coboundary operation is simply the linear map given by multiplication by $e_0 + e_1 + \dots e_n$, i.e. multiplication by the sum of all vertices!

None of this gives you an explicit example of a combinatorial Laplacian, but you can easily find one by finding the boundary operation matrix for your favorite simplicial complex, taking its transpose, and multiplying them together in the prescribed way(s).