Shelling (topology)

Last updated August 27, 2024

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, for all $k=2,3,\ldots$ , the complex

B_{k}:={\Big (}\bigcup _{i=1}^{k-1}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 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.^[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]

Notes

↑ Björner, Anders (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.
↑ Bruggesser, H.; Mani, P. "Shellable Decompositions of Cells and Spheres". Mathematica Scandinavica. 29: 197–205. doi: 10.7146/math.scand.a-11045 .
↑ Ziegler, Günter M. "8.2. Shelling polytopes". Lectures on polytopes. Springer. pp. 239–246. doi: 10.1007/978-1-4613-8431-1_8 .
↑ Rudin, Mary Ellen (1958). "An unshellable triangulation of a tetrahedron". Bulletin of the American Mathematical Society . 64 (3): 90–91. doi: 10.1090/s0002-9904-1958-10168-8 . ISSN 1088-9485.

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References

Kozlov, Dmitry (2008). Combinatorial Algebraic Topology. Berlin: Springer. ISBN 978-3-540-71961-8.

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[1] Björner, Anders (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] Bruggesser, H.; Mani, P. "Shellable Decompositions of Cells and Spheres". Mathematica Scandinavica. 29: 197–205. doi: 10.7146/math.scand.a-11045 .

[3] Ziegler, Günter M. "8.2. Shelling polytopes". Lectures on polytopes. Springer. pp. 239–246. doi: 10.1007/978-1-4613-8431-1_8 .

[4] Rudin, Mary Ellen (1958). "An unshellable triangulation of a tetrahedron". Bulletin of the American Mathematical Society . 64 (3): 90–91. doi: 10.1090/s0002-9904-1958-10168-8 . ISSN 1088-9485.

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