# Simplices

As a tool for measuring “holes” in the next section, we will need to construct $${n}$$-dimensional surfaces within a space. We will build these surfaces out of simple triangular pieces called simplices. The idea is to map triangles to the space, then build surfaces out of these triangles within the space.

• Standard n-simplex $${\Delta^{n}}$$: the $${n}$$-dimensional space defined by the join of the $${(n+1)}$$ unit points in $${\mathbb{R}^{n+1}}$$
• Singular n-simplex (AKA $${n}$$-simplex): a mapping $${\sigma\colon\Delta^{n}\to X}$$ of the standard $${n}$$-simplex into $${X}$$, the space being studied; the map is only assumed to be continuous (and so may be singular)
• n-chain: an element of $${C_{n}(X)}$$, defined to be the free abelian group with basis the n-simplices $${\sigma_{\alpha}}$$; i.e. the abelian group of finite formal linear sums (“chains”) $${\sum a_{\alpha}\sigma_{\alpha}}$$ with coefficients $${a_{\alpha}\in\mathbb{Z}}$$
• Boundary homomorphisms $${\partial_{n}\colon C_{n}\left(X\right)\to C_{n-1}\left(X\right)}$$: takes an $${n}$$-chain to the $${(n-1)}$$-chain consisting of the oriented sum of boundaries; also denoted simply $${\partial}$$

By oriented sum, it is meant that we desire a coherent orientation in $${X}$$; i.e. the boundary of a boundary should vanish. We can achieve this in any dimension by taking the boundaries $${\partial\Delta^{n}}$$ as oriented according to vertex order, and then defining $${\partial\sigma\left(\partial\Delta^{n}\right)\equiv\sum\left(-1\right)^{i}\sigma\left[\Delta^{n}-v_{i}\right]}$$. This formalism has the effect of reversing the “backwards” boundaries by preceding them with a minus sign, as can be seen with the heavier weight arrows in the preceding figure for a 2-simplex. For 0-simplices, we define $${\partial_{0}\sigma=0}$$.