Shelling (topology)

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

${\displaystyle B_{k}:={\Big (}\bigcup _{i=1}^{k-1}C_{i}{\Big )}\cap C_{k}}$

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.

• A shellable complex is homotopy equivalent to a wedge sum of spheres, one for each spanning simplex 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.

• Every Coxeter complex, and more generally every building (in the sense of Tits), is shellable.[1]

• The boundary complex of a (convex) polytope is shellable.[2][3] Note that here, shellability is generalized to the case of polyhedral complexes (that are not necessarily simplicial).

• There is an unshellable triangulation of the tetrahedron.[4]

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