Homotopy type of non-Cohen-Macaulay complexes

Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of same dimension). I would like to know examples of pure simplicial complexes (whose simplices are combinatorially defined) that are not shellable. Moreover, can anybody point out papers in which author(s) deal with simplcial complexes that do not have homotopy type of a wedge spheres?

An interesting class of pure complexes which are not Cohen-Macaulay are the chessboard complexes. See http://www.math.miami.edu/~wachs/papers/tmc.pdf. Another interesting class of non-Cohen-Macaulay complexes are the $$h$$-complexes of Edelman and Reiner. See https://arxiv.org/pdf/math/0311271.pdf.

In a recent article Minimal Cohen-Macaulay Simplicial Complexes by Dao, Doolittle, and Lyle many examples of CM complexes (which happen to be minimal by their definition) are listed in Section 5. For example, included in the list are triangulations of $$\mathbb{RP}^{2n}$$ or the dunce hat. Some other interesting examples are A non-partitionable Cohen-Macaulay simplicial complex and A balanced non-partitionable Cohen-Macaulay complex which are recent counterexamples to the "partionability conjecture."

Edit: Since the question asks about both shellablitly and CM-ness it is perhaps worth noting that shellablity depends of the particular simplicial complex and not just the geometric realization. So, all spheres and balls are CM, but there are non-shellable examples of each.

The order complex of a poset is the simplicial complex whose faces are the chains in the poset. So order complexes of combinatorially-defined posets are of the combinatorial flavor that you want, and the resulting complex is pure if and only if the poset is graded. But there are all kinds of examples of non-Cohen-Macaulay posets with natural definitions.

One typical kind of bad example is by subword or pattern containment. A particular instance of this is:

Sagan, Bruce E.; Vatter, Vincent, The Möbius function of a composition poset, J. Algebr. Comb. 24, No. 2, 117-136 (2006). ZBL1099.68081.

In this paper they look at all words of integers, ordered by subword containment. I guess that's graded by word length. Although the poset is infinite, intervals are finite, and typically are not Cohen-Macaulay.

Similar study has been made of permutation pattern containment by Peter McNamara, Jason Smith, Einar Steingrímsson and others. And here too, intervals often fail to be Cohen-Macaulay.

Every finite simplicial complex has the homotopy type of a triangulable manifold with boundary, and triangulated manifolds are pure. So the condition `pure' doesn't restrict the possible homotopy types for finite simplicial complexes.

• The pureness requirement is relevant for the shellability question, though. – Sam Hopkins Jan 24 '20 at 12:02
• Since every shellable complex is homotopy equivalent to a wedge of spheres, if we start with any finite simplicial complex that is not homotopy equivalent to a wedge of spheres (e.g., any complex with a non-trivial cup product in its cohomology) then we can replace it by a triangulated manifold that is pure but not shellable. – IJL Jan 24 '20 at 15:26
• yes but I think the original question asker was most interested in an example where the faces correspond to some reasonable combinatorial objects and containment is some easily described relationship between these objects. This situation often arises in algebraic combinatorics, and often the "goal" is to prove shellability/Cohen-Macaulayness. – Sam Hopkins Jan 24 '20 at 15:30
• I see your point; I'm not a combinatoricist algebraic or otherwise, so I was paying less attention to that part of the question. – IJL Jan 24 '20 at 15:32

Here are some examples: There are many papers that deal with topological properties of these complexes and they are rarely Cohen-Macaulay.

1) Triangulated manifolds are very interesting combinatorial objects, and usually, unless the manifold is a sphere or a ball they are not Cohen-Macaulay. Many aspects of the enumerative theory of face numbers of spheres have interesting generalization to manifolds. There is also rich examples of triangulations of specific manifolds with few vertices.

2) Within triangulations of manifolds there is a special role to $$d/2+1$$ neighborly manifolds when $$d$$ is even. Their existence for $$d=2$$ is the Heawood conjecture. For larger values of $$d$$ only a few examples are known like the Kuhnel 9-vertex triangulation of $$CP^2$$.

3) Triangulated pseudomanifolds (every co-dimension 1 face is included in two maximal faces + some connectivity assumption) can also be fascinating combinatorial objects.

Here are three more interesting examples.

4) Triangulations of algebraic varieties (with singularity) are very interesting combinatorial objects.

5) Quotients of buildings.

6) High dimensional expanders.