mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a shelling of a

simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

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

\Delta

be a finite or countably infinite simplicial complex. An ordering

C_1,C_2,\ldots

of the maximal simplices of

\Delta

is a shelling if, for all

k=2,3,\ldots

, the complex :

B_k:=\Big(\bigcup_^C_i\Big)\cap C_k

is pure and of dimension one smaller than

\dim C_k

. That is, the "new" simplex

C_k

meets the previous simplices along some union

B_k

of top-dimensional simplices of the boundary of

C_k

. If

B_k

is the entire boundary of

C_k

then

C_k

is called spanning. For

\Delta

not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of

\Delta

having analogous properties.

Properties

* A shellable complex is

homotopy equivalent In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

to a

wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the ...

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.

Examples

* Every Coxeter complex, and more generally every

building A building or edifice is an enclosed Structure#Load-bearing, structure with a roof, walls and window, windows, usually standing permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, a ...

(in the sense of Tits), is shellable. * The boundary complex of a (convex) polytope is shellable. Note that here, shellability is generalized to the case of polyhedral complexes (that are not necessarily simplicial). * There is an unshellable

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

of the

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

Notes

References

* {{cite book, first=Dmitry, last= Kozlov , title=Combinatorial Algebraic Topology , publisher=Springer , location=Berlin , year=2008 , isbn=978-3-540-71961-8 Algebraic topology Properties of topological spaces Topology