Let $G$ be a finite group with a BN-pair of rank $n$. Let $B$ be the associated Borel subgroup.

Let $P$ be the poset of *proper* right cosets (i.e. $Kg$ with $K \in [B,G)$ and $g \in G$).

Let $\hat{P}:= P \cup \{\hat{0}, \hat{1}\}$ be the bounded extension of $P$.

All the following questions are very closely related:

*Question 1.1:* Is $\hat{P}$ Cohen-Macaulay?

*Question 1.2:* Does $\hat{P}$ admits a dual EL-labeling?

*Question 2.1:* Is the order complex $\Delta(P)$ equal (or equivalent) to the building of the BN-pair (as an abstract simplicial complex)?

*Question 2.2:* Is the geometric realization $|\Delta(P)|$ homotopic to a wedge of $r \ge 1$ spheres $\mathbb{S}^{n-1}$.

*Question 2.3*: What is $r$? How to compute it?

**Edit (17/11/2016)**

We are not expert on these questions, but we are also not completely ignorant. All these questions (except perhaps Questions 1.2 and 2.3) should be obvious to experts, and we need their clarification.

*About Question 2.1*:

The definition of the building associated to a BN-pair (as written in [2, Chapter V], [3, Section 15.5] and [4, Section 5.7]) looks like to the definition of $\Delta(P)$ in [6, Section 1.1], but we are not sure that they are equivalent. One doubt comes from the fact that $\Delta(P)$ is defined from the poset of *proper* cosets, whereas, in [2] and [3], the building seems to be defined for the poset of all the cosets (in [4], it is with proper cosets). Anyway, is the equivalence immediate, or does it need a proof? Any reference?

*About Question 1.1*:

In [2, Remark 3 p94] it is written: << If you know what a Cohen-Macaulay complex is, then you can easily deduce from our study of the homotopy type of a building that every building is a Cohen-Macaulay complex. >> This result is written without proof, so it should be obvious to expert also. Anyway, is it explicitly proved in some other reference?

We are using the definition of Cohen-Macaulay poset of [7]: it is a property of the order complex of all the open intervals of the poset. Assuming that Question 2.1 has a positive answer, then, $\Delta(P)$ is a building, so is a Cohen-Macaulay *complex*. Can we deduce that $\hat{P}$ is a Cohen-Macaulay *poset*?

*About Question 1.2*:

It is known that a poset admitting a (dual) EL-labeling is Cohen-Macaulay, but the converse is false. So this question seems relevant. Is it an open problem?

*About Question 2.2*:

By [2, Theorem 2 p 93], it has a positive answer, if Question 2.1 has so.

*About Question 2.3*:

Let $G$ be a semisimple algebraic group defined over some field $k$. In [5, Section 4, (ii')], Tits states that << A spherical building has the homotopy type of a bouquet of spheres.
Furthermore, the number $N$ of these spheres is easily determined; for instance, if the ground field $k$ is finite of characteristic $p$, $N$ is the $p$-contribution to the order of $G$. >>

Is there a proof of this statement somewhere? Is it true for any finite group with a BN-pair? Or should we assume $G$ of Lie type? and if so how to compute $r$ (denoted $N$ by Tits) beyond the Lie type?

**References**

$[1]$ Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics, 2002.

$[2]$ Kenneth S. Brown, Buildings, 1989.

$[3]$ Roger W. Carter, Simple groups of Lie type, 1972.

$[4]$ Paul Garrett, Buildings and classical groups, 1997.

$[5]$ J. Tits, On buildings and their applications, Proceedings of the ICM (Vancouver), 1975.

$[6]$ Michelle L. Wachs, Poset topology: tools and applications, 2007.

$[7]$ Anders Björner, Adriano M. Garsia, and Richard P. Stanley, An introduction to Cohen-Macaulay partially ordered sets, 1982.

**Edit (18/11/2016)**

*About Question 2.1*:

According to the notation of [Section 1.1, 6], from a simplicial complex $\Delta_0$, we can make a poset $P(\Delta_0)$ called the face poset of $\Delta_0$, and from a poset $P_0$, we can make a simplicial complex $\Delta(P_0)$ called the order complex of $P_0$. The result is (stated without proof in [Section 1.1, 6]): $$|\Delta_0| \simeq |\Delta(P(\Delta_0))|$$ i.e. the geometric realization are homeomorphic. Now take $[H,G]$ a boolean interval of finite groups, $P_0:=C(H,G)$ the poset of proper right cosets $Kg$, and $P_0^*$ the dual of $P_0$ (reversed order). Then $P_0^*$ is naturally the face poset of a simplicial complex, which can be called $P^{-1}(P_0^*)$. If $H$ is also a Borel subgroup coming from a BN-pair, then $P^{-1}(P_0^*)$ is the simplicial complex structure of the building associated to the BN-pair. By applying the result above, it follows that $$|P^{-1}(P_0^*)| \simeq |\Delta(P(P^{-1}(P_0^*)))| = |\Delta(P_0^*)| \simeq |\Delta(P_0)|$$
The fact that some references consider all the cosets whereas other references consider only proper cosets, does not matter because $G$ would map to the empty simplex.

So we have a positive answer to Question 2.1.

**Edit (19/11/2016)**

*About Question 1.1*:

Russ Woodroofe has points out to us the reference answering this question:

Anders Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math. 52 (1984), no. 3, 173–212.

So now, all the questions have been answered, except Questions 1.2 and 2.3 (in general).

dualEL-labeling is (I am clearly not an expert for these things) -- does that just mean that the dual complex has an EL-labeling? In any case, it is known that thin buildings of classic type (i.e. apartments) are EL-shellable. Also, building are shellable (as shown in Björner's paper pointed out to you by Russ Woodroofe). (Correction: I gave the wrong reference for the EL-shellability of apartments) $\endgroup$ – Max Horn Nov 19 '16 at 20:20