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 of dimension ${\displaystyle \dim C_{k}-1}$ 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.

Examples

• Every Coxeter complex, and more generally every building, is shellable.[1]

• There is an unshellable triangulation of the tetrahedron.[2]

Notes

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, Mary Ellen (1958-02-14). "An unshellable triangulation of a tetrahedron". Bulletin of the American Mathematical Society. 64 (3): 90–91. doi:10.1090/s0002-9904-1958-10168-8. ISSN 1088-9485.

References

• Kozlov, Dmitry (2008). Combinatorial Algebraic Topology. Berlin: Springer. ISBN 978-3-540-71961-8.