Simplex - Algebraic Topology

Algebraic Topology

In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.

A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with 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 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 face of an n-simplex is an affine n-1-simplex, and thus the boundary of an n-simplex is an affine n-1-chain. Thus, if we denote one positively-oriented affine simplex as

with the denoting the vertices, then the boundary of σ is the chain

$\partial\sigma = \sum_{j=0}^n (-1)^j$ .

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:

$\partial^2\sigma = \partial ( ~ \sum_{j=0}^n (-1)^j ~ ) =0.$

Likewise, the boundary of the boundary of a chain is zero: .

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map . In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,

where the are the integers denoting orientation and multiplicity. For the boundary operator, one has:

where 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 map operation (by definition of a map).

A continuous map to a topological space X is frequently referred to as a singular n-simplex.

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