Shelling (topology)

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 \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:=\left(\bigcup_{i=1}^{k-1}C_i\right)\cap C_k is pure and (\dim C_k-1)-dimensional 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

Examples

References

  1. Björner, Anders (June 1984). "Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings". Advances in Mathematics 52 (3): 173–212. doi:10.1016/0001-8708(84)90021-5. ISSN 0001-8708.
  2. Rudin, M.E. (1958-02-14). "An unshellable triangulation of a tetrahedron". Bull. Am. Math. Soc. 64 (3): 90–91. doi:10.1090/s0002-9904-1958-10168-8. ISSN 1088-9485.
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