# How is the sum of simplices defined?

I have recently started to learn about simplices, simplicial complexes and simplicial homology. I understand that for some set of points $$\{ p_0, \ldots, p_k \} \subset \mathbb{R}^n$$ a simplex $$\sigma$$ is the set

$$\sigma \enspace \equiv \enspace < \; p_0, \ldots, p_k \; > \enspace := \enspace \Big\{ \; x \in \mathbb{R}^n \; \Big | \; x = \sum_{i = 0}^k \lambda_i p_i \, , \; \lambda_i > 0 \, , \; \sum_{i = 0}^k \lambda_i = 1 \; \Big\}$$

Now, the author of my script started adding and substracting simplices, e.g. when defining the boundary of $$\sigma$$ he writes

$$\partial_k \sigma \enspace = \enspace < \; p_1, \ldots, p_k \; > - < \; p_0, p_2, \ldots, p_k \; > + \ldots + < \; p_0, \ldots, p_{k-1} \; >$$

However, he did not define what the operations $$+$$ and $$-$$ mean in this context. I also seem to have trouble finding a rigorous definition of those operations for simplices on the web. Can anyone explain?

• It is a formal sum. You treat simplices as symbols to be added and subtracted and manipulate the sums according to the usual rules for addition and subtraction. Commented Feb 22, 2021 at 22:49
• So the resulting object is not really defined? Does that mean that there is also no group operation explicitly defined for the chain groups? Commented Feb 22, 2021 at 22:58
• No, it's formal. Have you seen the notion of a free vector space or free abelian group? Commented Feb 22, 2021 at 23:35

The short answer is: the maps $$\partial$$ are defined on the free abelian groups generated by $$n$$-simplices, hence it makes sense to speak about sums of simplices in these groups.

I'll try to give a down to earth answer, for which we will need to detour first. Consider the set $$\mathbb{Z}[X]$$ of integer polynomials. An element there is an expression $$a_0 + a_1 X + \ldots + a_nX^n$$ where each $$a_i$$ is an integer, and $$X$$ is an indeterminate. Even though we might be quite familiar with thinking of polynomials in this way, the terms 'expression' and 'indeterminate' are informal. A rigorous definition of $$\mathbb{Z}[X]$$ could be the set of sequences of integers that are eventually zero,

$$\mathbb{Z}[X] = \{(a_n)_{n \geq 0} \in \mathbb{Z}^{\mathbb{N}_0} : \text{there exists k \in \mathbb{N}_0 such that a_n = 0 for all k \geq n}\}. \tag{1}$$

Given integers $$a_1,\ldots,a_n\in \mathbb{Z}$$, we then define

$$a_0 + a_1 X + \ldots + a_nX^n := (a_0,a_1,\ldots,a_n,0,0,\ldots) \tag{2}.$$

All the operations on polynomials can be defined in terms of sequences as in $$(1)$$ and are compatible with the usual notation $$(2)$$. This is a way to formally capture the idea of having some unknown element $$X$$ that can be 'scaled' by integers, multiplied by itself, and such that we can make sense of the sums of expressions like this.

In the same spirit, suppose you have a set $$X$$ and you want to have another set $$M$$ such that:

• $$X$$ is, in some sense, contained in $$M$$,
• we can make sense of multiplying $$x \in X$$ by an integer $$k \in \mathbb{Z}$$,
• we can give meaning to an expression like $$17x+28y+3z$$ for some $$x,y,z \in X$$.

A way to do this is to define the free abelian group $$\mathbb{Z}^{(X)}$$ with basis $$X$$. This concept is closely related to bases of vector spaces. The definition (technically, one definition, there are 'many' equivalent ones) is as follows: as a set $$\mathbb{Z}^{(X)}$$ consists of functions $$f \colon X \to \mathbb{Z}$$ such that $$f(x) \neq 0$$ for finitely many $$x \in X$$. In other words, the elements of $$\mathbb{Z}^{(X)}$$ are finitely supported functions $$\mathbb{Z} \to X$$. We can make sense of sum and multiplication of elements here in the same way we do to define a vector space structure on $$\mathbb{R}^X$$. Namely:

• the sum of $$f,g \in \mathbb{Z}^{(X)}$$ is the function $$(f+g)(x) := f(x)+g(x)$$.
• if $$k$$ is an integer and $$f \in \mathbb{Z}^{(X)}$$, we define $$(k\cdot f)(x) := kf(x)$$.

You can check that these operations define once again elements of $$\mathbb{Z}^{(X)}$$.

Now, let's go back to the analogy with polynomials; in particular, to the relation between definition $$(1)$$ and notation $$(2)$$. Even though formally the free abelian group is defined as finitely supported functions, we want to think of it as "$$\mathbb{Z}$$-linear combinations of $$X$$". To do this, we write $$x := \chi_{\{x\}}, \quad \chi_{\{x\}}(y) = \begin{cases}1 &\text{if x=y}\\ 0 &\text{otherwise}\end{cases}$$ You can check that, following this notation, every element of $$\mathbb{Z}^{(X)}$$ can be written as a finite sum $$a_1 x_1 + \ldots +a_n x_n$$ for some $$x_i \in X$$ and integers $$a_i$$. Moreover, two such expressions are equal if the elements of $$X$$ appearing are the same, and the coefficients accompanying each element coincide.

In this case, we consider $$\mathbb{Z}^{(C_n)}$$ with $$C_n$$ the set of $$n$$-simplices. In $$\mathbb{Z}^{(C_n)}$$ it makes perfect sense to speak of $$3 \sigma + 22 \tau$$ for some pair of simplices $$\sigma,\tau$$ or any expression of this sort; likewise the maps $$\partial_k$$ are well defined.

I hope this gives a little bit more context to the first sentence of this answer.