Editing Simplex (section)

== Algebraic topology ==
<!--'Singular n-simplex' redirects here-->
In [[algebraic topology]], simplices are used as building blocks to construct an interesting class of [[topological space]]s called [[simplicial complex]]es. These spaces are built from simplices glued together in a [[combinatorics|combinatorial]] fashion. Simplicial complexes are used to define a certain kind of [[homology (mathematics)|homology]] called [[simplicial homology]].

A finite set of {{mvar|k}}-simplexes embedded in an [[open subset]] of {{math|'''R'''<sup>''n''</sup>}} is called an affine {{mvar|k}}-[[Chain (algebraic topology)|chain]].  The simplexes in a chain need not be unique; they may occur with [[Multiplicity (mathematics)|multiplicity]]. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite [[orientability|orientation]], these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

Note that each facet of an {{mvar|n}}-simplex is an affine {{math|(''n'' − 1)}}-simplex, and thus the [[boundary (topology)|boundary]] of an {{mvar|n}}-simplex is an affine {{math|(''n'' − 1)}}-chain. Thus, if we denote one positively oriented affine simplex as
: <math>\sigma=[v_0,v_1,v_2,\ldots,v_n]</math>
with the <math>v_j</math> denoting the vertices, then the boundary <math>\partial\sigma</math> of {{mvar|σ}} is the chain
: <math>\partial\sigma = \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n].</math>

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
: <math>\partial^2\sigma = \partial \left( \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n] \right) = 0. </math>

Likewise, the boundary of the boundary of a chain is zero: <math> \partial ^2 \rho =0 </math>.

More generally, a simplex (and a chain) can be embedded into a [[manifold]] by means of smooth, differentiable map <math>f:\mathbf{R}^n \to M</math>. In this case, both the summation convention for denoting the set, and the boundary operation commute with the [[embedding]]. That is,
: <math>f \left(\sum\nolimits_i a_i \sigma_i \right) = \sum\nolimits_i a_i f(\sigma_i)</math>
where the <math>a_i</math> are the integers denoting orientation and multiplicity. For the boundary operator <math>\partial</math>, one has:
: <math>\partial f(\rho) = f (\partial \rho)</math>
where {{mvar|ρ}} is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the [[function (mathematics)|map operation]] (by definition of a map).

A [[continuous function (topology)|continuous map]] <math>f: \sigma \to X</math> to a [[topological space]] {{mvar|X}} is frequently referred to as a '''singular {{mvar|n}}-simplex'''<!--boldface per WP:R#PLA-->. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)<ref>{{cite book |first=John M. |last=Lee |title=Introduction to Topological Manifolds |url=https://books.google.com/books?id=AdIRBwAAQBAJ&pg=PR1 |date=2006 |publisher=Springer |isbn=978-0-387-22727-6 |pages=292–3}}</ref>