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.

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, for all , the complex

is pure and of dimension one smaller than . 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.

• 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.