# Shelling (topology)

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

|CitationClass=book }}