# Cycles in Tits building

Tits building for an $$n$$-dimensional vector space $$V$$ is defined to be the simplicial complex corresponding to the poset of proper and non-zero subspaces of $$V$$. It is denoted by $$T(V)$$. This is known to have the homotopy type of $$n-2$$-spheres. Any choice of linearly independent lines in $$V$$, define a sphere in $$V$$. The Steinberg module is defined as $$St=H_{n-2}(T(V))$$. Often for the choice of linearly independent vectors $$v_1,\ldots v_n$$ in $$V$$, we denote the corresponding sphere in the Steinberg module by $$[v_1,\ldots ,v_n]$$. These symbols satisfy certain properties for example multiplying each vector by a scalar doesn't change it, or switching two vectors multiplies the symbol by $$-1$$. Especially these symbols generate the Steinberg module. There are some non-trivial relations between these symbols. (We can get rid of them if we choose symbols that correspond to upper-triangular matrices, after a choice of a basis for $$V$$.)

We choose a subset $$S$$ of vertices of $$T(V)$$ i.e. non-zero and proper subspaces of $$V$$. Let's color the vertices in $$S$$ by red color. We call an $$n-2$$ cycle in $$T(V)$$ to be red iff all of its vertices lie in $$S$$. We call a homology class of the form $$[v_1,\ldots , v_n]$$ a red sphere iff all vertices corresponding to it i.e. all subspaces spanned by a subset of $$v_i$$'s are in $$S$$. My question is the following:

Is every red cycle a linear combination of red spheres?

The question is equivalent to the following one: Is there a representation for every $$n-2$$ homology class of $$T(V)$$ in terms of $$[v_1,\ldots , v_n]$$, such that none of the vertices of the symbols involved in this representation get cancelled and have an image in the final homology class.