Shelling (topology)

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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.


A d-dimensional simplicial complex is called pure if its maximal simplices all have dimension d. Let be a finite or countably infinite simplicial complex. An ordering of the maximal simplices of is a shelling if the complex is pure and -dimensional for all . That is, the "new" simplex meets the previous simplices along some union of top-dimensional simplices of the boundary of . If is the entire boundary of then is called spanning.

For not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of having analogous properties.


  • A shellable complex is homotopy equivalent to a wedge sum of spheres, one for each spanning simplex and 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.



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