I'm studying Jakob Jonsson's book Simplicial Complexes of Graphs very rigorously and in depth. I've been okay so far with the intensity and notation, but on page Chapter 3, section 2, page 30, I'm coming across some difficulty with his notation.
He writes:
"Let $\Delta$ be a simplicial complex. For $d \geq -1$, let $\tilde{C}_{d}(\Delta; \mathbb{F})$ be the free $\mathbb{F}$-module with one basis element, denoted as $[\sigma] = [s_1] \wedge \cdots \wedge [s_{d+1}]$, for each $d$-dimensional face $\{s_1, \dots, s_{d+1}\}$ of $\Delta$."
Later, he says:
"Whenever $\sigma$ and $\tau$ are disjoint faces such that $\sigma \cup \tau \in \Delta$, we define $[\sigma] \wedge [\tau]$ in the natural manner."
First question: What is this "natural manner" that he means?
Later, he defines reduced homology in its typical setting. He then gives an example:
"... note that $\tilde{H}_{d}(\Delta; \mathbb{F}) = 0$ for all $d$ whenever $\Delta = \text{Cone}_{x}(\Sigma)$ for some simplicial complex $\Sigma$. Namely, we may write any element $c$ in $\tilde{C}_{d}(\Delta; \mathbb{F})$ as $c = [x] \wedge c_1 + c_2$, where $c_1$ and $c_2$ are elements in $\tilde{C}(\Sigma; \mathbb{F})$." (he doesn't specify the dimension of this aforementioned module, but I assume its codimension with respect to $\Delta$ is one).
My next question is: With his notation, I'm unsure how to write an element $c \in \tilde{C}_{d}(\Delta;\mathbb{F})$. I would be pleased to see an example what a typical element may look like. I'm assuming it's something like $c = a_{1}[\sigma_1] + \cdots + a_{d}[\sigma_d]$, where coefficients are in $\mathbb{F}$.
So, yeah, what's the natural manner of $\wedge$, and how does a typical element look?