In mathematics, a shelling of a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

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 the complex :

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

is pure and of dimension

\dim C_k-1

for all

k=2,3,\ldots

. 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, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...

to a

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spheres The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for th ...

, 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 In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apa ...

, and more generally every

building A building, or edifice, is an enclosed structure with a roof and walls standing more or less permanently in one place, such as a house or factory (although there's also portable buildings). Buildings come in a variety of sizes, shapes, and funct ...

(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 ...

of the

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

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