# 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 ${\displaystyle \Delta }$ be a finite or countably infinite simplicial complex. An ordering ${\displaystyle C_{1},C_{2},\ldots }$ of the maximal simplices of ${\displaystyle \Delta }$ is a shelling if the complex ${\displaystyle B_{k}:=\left(\bigcup _{i=1}^{k-1}C_{i}\right)\cap C_{k}}$ is pure and ${\displaystyle (\dim C_{k}-1)}$-dimensional for all ${\displaystyle k=2,3,\ldots }$. That is, the "new" simplex ${\displaystyle C_{k}}$ meets the previous simplices along some union ${\displaystyle B_{k}}$ of top-dimensional simplices of the boundary of ${\displaystyle C_{k}}$. If ${\displaystyle B_{k}}$ is the entire boundary of ${\displaystyle C_{k}}$ then ${\displaystyle C_{k}}$ is called spanning.

For ${\displaystyle \Delta }$ not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of ${\displaystyle \Delta }$ 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.

## References

1. ^ Björner, Anders (June 1984). "Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings". Advances in Mathematics. 52 (3): 173–212. doi:10.1016/0001-8708(84)90021-5. ISSN 0001-8708.
2. ^ Rudin, M.E. (1958-02-14). "An unshellable triangulation of a tetrahedron". Bull. Am. Math. Soc. 64 (3): 90–91. doi:10.1090/s0002-9904-1958-10168-8. ISSN 1088-9485.