{{Short description|Mathematical concept}}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 <math>\Delta</math> be a finite or countably infinite simplicial complex. An ordering <math>C_1,C_2,\ldots</math> of the maximal simplices of <math>\Delta</math> is a '''shelling''' if, for all <math>k=2,3,\ldots</math>, the complex :<math>B_k:=\Big(\bigcup_{i=1}^{k-1}C_i\Big)\cap C_k</math> is pure and of dimension one smaller than <math>\dim C_k</math>. That is, the "new" simplex <math>C_k</math> meets the previous simplices along some union <math>B_k</math> of top-dimensional simplices of the boundary of <math>C_k</math>. If <math>B_k</math> is the entire boundary of <math>C_k</math> then <math>C_k</math> is called '''spanning'''.
For <math>\Delta</math> not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of <math>\Delta</math> having analogous properties.
==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.
==Examples==
* Every Coxeter complex, and more generally every building (in the sense of Tits), is shellable.<ref>{{Cite journal | issn = 0001-8708 | volume = 52 | issue = 3 | pages = 173–212 | last = Björner | first = Anders | author-link = Anders Björner | title = Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings | journal = Advances in Mathematics | date = 1984 | doi = 10.1016/0001-8708(84)90021-5 | doi-access = free }}</ref>
* The boundary complex of a (convex) polytope is shellable.<ref>{{cite journal | last1=Bruggesser | first1=H. | last2=Mani | first2=P. | title=Shellable Decompositions of Cells and Spheres. | journal=Mathematica Scandinavica | date=1971 | volume=29 | pages=197–205 | doi=10.7146/math.scand.a-11045 | doi-access=free}}</ref><ref>{{cite book | last1=Ziegler | first1=Günter M. | authorlink1=Günter M. Ziegler | title=Lectures on polytopes | section=8.2. Shelling polytopes | pages=239–246 | publisher=Springer | doi=10.1007/978-1-4613-8431-1_8 | doi-access=free}}</ref> 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.<ref>{{Cite journal | issn = 1088-9485 | volume = 64 | issue = 3 | pages = 90–91 | last = Rudin | first = Mary Ellen | author-link = Mary Ellen Rudin | title = An unshellable triangulation of a tetrahedron | journal = Bulletin of the American Mathematical Society | date = 1958 | doi=10.1090/s0002-9904-1958-10168-8 | doi-access = free }}</ref>
==Notes==
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==References==
* {{cite book|first=Dmitry|last= Kozlov |title=Combinatorial Algebraic Topology |publisher=Springer |location=Berlin |year=2008 |isbn=978-3-540-71961-8}}
Category:Algebraic topology Category:Properties of topological spaces Category:Topology