# Shelling (topology): Difference between revisions

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==Definition== | ==Definition== | ||

A ''d''-dimensional simplicial complex is called '''pure''' if its maximal simplices all have dimension ''d''. Let <math>\Delta</math> be a finite or countably infinite simplicial complex. An ordering <math>C_1,C_2,\ldots</math> of the maximal simplices of <math>\Delta</math> is a '''shelling''' if the complex <math>B_k:=\left(\bigcup_{i=1}^{k-1}C_i\right)\cap C_k</math> is pure and <math>(\dim C_k-1)</math>-dimensional for all <math>k=2,3,\ldots</math>. If <math>B_k</math> is the entire boundary of <math>C_k</math> then <math>C_k</math> is called '''spanning'''. | A ''d''-dimensional simplicial complex is called '''pure''' if its maximal simplices all have dimension ''d''. Let <math>\Delta</math> be a finite or countably infinite simplicial complex. An ordering <math>C_1,C_2,\ldots</math> of the maximal simplices of <math>\Delta</math> is a '''shelling''' if the complex <math>B_k:=\left(\bigcup_{i=1}^{k-1}C_i\right)\cap C_k</math> is pure and <math>(\dim C_k-1)</math>-dimensional for all <math>k=2,3,\ldots</math>. That is, the "new" simplex <math>C_k</math> meets the previous simplices along some union <math>B_k</math> of top-dimensional simplices of the boundary of <math>C_k</math>. If <math>B_k</math> is the entire boundary of <math>C_k</math> then <math>C_k</math> is called '''spanning'''. | ||

For <math>\Delta</math> not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of <math>\Delta</math> having analogous properties. | For <math>\Delta</math> not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of <math>\Delta</math> having analogous properties. |

## Revision as of 15:49, 28 February 2014

In mathematics, a **shelling** of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitting a shelling is called **shellable**.

## Definition

A *d*-dimensional simplicial complex is called **pure** if its maximal simplices all have dimension *d*. Let be a finite or countably infinite simplicial complex. An ordering of the maximal simplices of is a **shelling** if the complex is pure and -dimensional for all . That is, the "new" simplex meets the previous simplices along some union of top-dimensional simplices of the boundary of . If is the entire boundary of then is called **spanning**.

For not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of having analogous properties.

## Properties

- A shellable complex is homotopy equivalent to a wedge sum of spheres, one for each spanning simplex and of corresponding dimension.
- A shellable complex may admit many different shellings, but the number of spanning simplices, and their dimensions, do not depend on the choice of shelling. This follows from the previous property.

## Examples

- Every Coxeter complex, and more generally every building, is shellable.
^{[1]}

- There is an unshellable triangulation of the tetrahedron.
^{[2]}

## References

- {{#invoke:citation/CS1|citation

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