Skeleton (topology)

In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace X_n that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step.

These subspaces increase with n. The 0-skeleton is a discrete space, and the 1-skeleton a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when X has infinite dimension, in the sense that the X_n do not become constant as n → ∞.

In geometry, a k-skeleton of n-polytope P (functionally represented as skel_k(P)) consists of all i-polytope elements of dimension up to k.

For example:

The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set ${\displaystyle K_{*}}$ can be described by a collection of sets ${\displaystyle K_{i},\ i\geq 0}$, together with face and degeneracy maps between them satisfying a number of equations. The idea of the n-skeleton ${\displaystyle sk_{n}(K_{*})}$ is to first discard the sets ${\displaystyle K_{i}}$ with ${\displaystyle i>n}$ and then to complete the collection of the ${\displaystyle K_{i}}$ with ${\displaystyle i\leq n}$ to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees ${\displaystyle i>n}$.

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