Polyhedral Subdivisions and Projections of Polytopes Jörg Rambau Dissertation (Advisor: Günter M. Ziegler, TU Berlin)

   
Combinatorial Models

When dealing with topological problems, it is often useful to work with special combinatorial objects related to the topological spaces under consideration, rather than with the topological spaces themselves. For example,   triangulations of topological spaces lead to simplicial complexes that carry the complete topological information about the original space within their combinatorial structure. There are two quite different links between the Generalized Baues Conjecture and the theory of combinatorial models of topological spaces: loop spaces and finite-dimensional Grassmannians.

  Combinatorial models of the finite-dimensional Grassmannians are closely related to oriented matroids. The   real Grassmann manifold $G_k(\mathbb{R} ^d)$, or   Grassmannian for short, is the space of all $k$-dimensional subspaces of $\mathbb{R} ^d$ with the usual quotient topology, where $V \sim W$ if and only if $\lin(V) = \lin(W)$ for $V = (v_1, \dots , v_k)$ and $W = (w_1, \dots , w_k)$in $(\mathbb{R} ^d)^k$. The oriented matroids $\mathcal{M}(V)$ and $\mathcal{M}(W)$ of equivalent representatives $V \sim W$ coincide. Thus, the set of all   realizable oriented matroids on $d$ points of rank $k$ with the topology inherited by the weak map relation can be regarded as a model of $G_k(\mathbb{R} ^d)$. But the realizability assumption disturbs the combinatorial shape of the model; there is no combinatorial criterion known to check realizability efficiently. (For additional problems occurring in this stratification see STURMFELS [69].)

So a natural idea is to bring all oriented matroids on $d$ points of rank $k$ -- realizable or not -- into the game, forming the   MacPhersonian $MacP(d,k)$. This was actually done by GELFAND & MACPHERSON [30] who conjectured that $MacP(d,k)$ is homotopy equivalent to $G_k(\mathbb{R} ^d)$, which is in fact true if $k \le 3$ (see BABSON [3] and MNËV & ZIEGLER [54]).

A generalization of this concept is the OM-Grassmannian (we refer to BABSON [3], MNËV & ZIEGLER [54], and RICHTER-GEBERT [63, Introduction, Section 4]).

Definition 1.2.15   Let $\mathcal{M}^d$ be an oriented matroid of rank $d$ on the set $[n]$. The   OM-Grassmannian of $\mathcal{M}$ is the poset

$$\mathcal{G}_k(\mathcal{M}^d) := \setof{\mathcal{N}^k}{ \tex... ...space is a rank $k$\space strong image of $\mathcal{M}^d$ }},$$

partially ordered by the weak map relation.

The unifying conjecture is the following.

Conjecture 1.2.16   If $\mathcal{M}^d$ is realizable, then the   order complex of its OM-Grassmannian $\mathcal{G}_k(\mathcal{M}^d)$ is homotopy equivalent to the real Grassmannian $G_k(\mathbb{R} ^d)$ for all $1 \le k < n$.

For $k \le 2$ there are affirmative results. For $\mathcal{M}^d = \mathcal{F}^d$ we get the MacPhersonian $MacP(d,k)$. In the case $k = d - 1$ we are again concerned with the extension space conjecture, inheriting the corresponding partial results. Further affirmative partial answers to the extension space conjecture would support inductive approaches to this problem.

The   model theory of loop spaces deals with the construction of combinatorial models for the loop space $\Omega X$ of $X$, which is the space of all closed paths in a topological space $X$, endowed with a certain topology. The source of the Generalized Baues Conjecture actually lies in this field. The original Baues Conjecture belongs to the purely combinatorial part of a whole theory developed by -- among others -- ADAMS [1], MILGRAM [51], and BAUES [5]. The exact setting of this framework requires much more insight into category theory than we can present here. Thus, we restrict ourselves to a sketch of the situation.

In the book The Geometry of Loop Spaces by BAUES [5] a general model theorem is presented, which roughly states the following: If $X$ is a space glued together from standard building blocks, then, under certain conditions, its loop space $\Omega X$ is glued together from certain path spaces of the standard building blocks. The following version of the model theorem is still a rough sketch of the exact setting.

 

Theorem 1.2.17 (Model Theorem for Loop Spaces, sketch)   (BAUES [5]) Let $\mathcal{K}$ be a collection of abstract objects $K$ with the following properties.

(i)

 Every $K \in \mathcal{K}$ has a facial structure,

(ii)

  for every $K \in \mathcal{K}$ all faces of $K$ are again in $\mathcal{K}$ and have the inherited facial structure,

(iii)

  every $K \in \mathcal{K}$ is a face of another $K'$ in $\mathcal{K}$ and has the inherited facial structure,

(iv)

  for every $K \in \mathcal{K}$ there is a topological standard realization $\rho(K)$ of $K$ that is homeomorphic to a ball,

(v)

  there is another collection $\Omega \mathcal{K}$ of abstract objects satisfying i, ii, and iii with the following property: for every $K \in \mathcal{K}$ there is an object $\Omega K$ and a corresponding topological standard realization $L(\Omega K)$ that is (weakly) homotopy equivalent to a certain path space $\Omega \rho (K)$ of $\rho(K)$,

(vi)

  for every $K \in \mathcal{K}$ the boundary $\partial L(\Omega K)$ is homotopy equivalent to a sphere of appropriate dimension.

Let $X$ be a topological space that is the topological realization $[C]_{\rho}$ of a complex $C$ consisting of building blocks $K$ from  $\mathcal{K}$. Then there is a construction that builds up a complex $\Omega C$ from the building blocks $\Omega K$ in $\Omega \mathcal{K}$ such that the topological realization $[\Omega C]_L$ of $\Omega C$ is (weakly) homotopy equivalent to the loop space $\Omega X$ of $X$. In other words, the following formula holds:

$$[\Omega C]_L \sim \Omega [C]_{\rho}.$$

Because a closer inspection of this construction is outside the scope of this thesis, we restrict ourselves to the problem of guaranteeing the assumptions of the theorem.

We briefly sketch how one gets $\Omega K$ for building blocks $K$ that are abstract contractible cell complexes containing at least two vertices. Let $\Omega (\rho(K),x_s,x_t)$ be the space of all paths in $\rho(K)$ starting at $x_s \in \rho(K)$ and terminating at $x_t \in \rho(K)$, where $x_s := \rho (s_K)$ and $x_t := \rho (t_K)$correspond to vertices $s_K$ and $t_K$ in $K$. This   path space is (via certain topological constructions) related to the loop space of $\rho(K)$. We construct a model $\Omega (K, s, t)$for $\Omega (\rho(K),x_s,x_t)$. To any element $F$ of $K$ we assign vertices $s(F)$ and $t(F)$ of $F$ with the following properties:

$$\begin{align*}s(K) &= s_K,\\ t(K) &= t_K,\\ s(F') &= s(F) \quad\text{if $F'$... ...quad\text{if $F'$\space is a face of $F$\space and $t(F) \in F'$ }. \end{align*}$$

The pair $(s,t)$ is called a   double stippling on $K$. A   cellular string in $(K,s,t)$ with respect to the double stippling $(s,t)$ is a sequence $(F_0, \dots , F_m)$ of faces of $K$ such that $s(F_i) = t(F_{i-1})$ for all $i = 0, \dots , m$. The set $\omega (K,s,t)$ of all cellular strings in $K$ is partially ordered by refinement, i. e.,

$$\begin{multline*}(F_0, \dots , F_m) \le (G_0, \dots , G_k) :\iff\\ \text{there... ...lular string in $G_{\nu}$\space for all $\nu = 0, \dots , k$ }. \end{multline*}$$

The   (cellular) string complex $\Omega K = \Omega (K,s,t)$(see Figure 1.10) is a certain abstract cell complex that induces this partial order via its facial relations. We may interpret this model as follows: for any path $w$ from $x_s$ to $x_t$ in $\rho(K)$ consider the string of those faces of $\rho(K)$ that are visited by $w$. This string gives rise to a cellular string in $K$, thus to a cell in $\Omega K$.

The crucial part in applications of the Model Theorem, as far as our framework is concerned, is the construction of the   spherical desuspensions $L(\Omega K)$. These are the standard realizations satisfying v and vi of Theorem 1.2.17. Constructing a spherical desuspension $L(\Omega K)$ roughly means finding a topological cell complex with spherical boundary that carries the combinatorial structure of $\Omega K$ within its incidences and is ``consistent'' for all $K$.

One straightforward attempt is to consider a topological realization of the   order complex of $\Omega K$, which is the first barycentric subdivision of $\Omega K$. Its boundary is just a topological realization of the order complex of $\Omega K \sm \{ K \}$. If this boundary is always homotopy eqivalent to a sphere, then we are done. For the special case of double-stipplings on a polytope coming from a generic linear functional this is exactly the problem that is addressed by the Generalized Baues Conjecture for $d' = 1$.

The following considerations lead to the original Baues Conjecture. If we are concerned with a triangulated topological space $X$, so that $\mathcal{K}$ is the set of all abstract simplices with standard realizations, then the string complex for the $n$-simplex $\Delta_n$ is combinatorially equivalent to an $(n-1)$-cube (see Figure 1.10). Hence, by the Model Theorem, the loop space $\Omega X$ of $X$ is (weakly) homotopy equivalent to a certain space glued together from cubes. Continuing this process with $\Omega X$ instead of $X$ leads to the string complex of the $n$-cube, which turns out to be the   $(n-1)$-permutahedron, the convex hull of all permutations of $[n-1]$, viewed as vectors in $\mathbb{R} ^{n-1}$.

  

Figure 1.10: The string space of the $3$-simplex stippled by minimal and maximal vertex. It is modelled by a $2$-cube. The indicated path going first through the face $\{ 1,2,3 \}$ and then along the edge $\{ 3,4 \}$ corresponds to the grey point in the relative interior of the edge $\{ (134),(1234) \}$ of the square.

The next step, however, yields string complexes that cannot be realized as polytopes. They even fail to be homeomorphic to balls. To apply the Model Theorem, however, it would be sufficient to come up with realizations whose boundary is homotopy equivalent to a sphere. The Baues Conjecture is roughly as follows.

 

Conjecture 1.2.18 (Baues Conjecture)   (BAUES [5]) The boundary of the string complex of a permutahedron has the homotopy type of a sphere.

Here is our connection: assume that $\rho(K)$ is a permutahedron with face lattice $K$, and the double stippling on $K$ is given by a generic linear functional $\psi$on $\rho(K)$. Then the first barycentric subdivision of the string complex $\Omega K$ coincides with the order complex of the poset of all $\psi$-induced subdivisions. The minimal and the maximal vertex of $\rho(K)$ with respect to $\psi$ correspond to the fixed base points in $\rho(K)$. Thus, the proof of the case $d' = 1$ of the Generalized Baues Conjecture by BILLERA, KAPRANOV & STURMFELS [9] implies the Baues Conjecture.

Our negative result in Chapter 2 can be regarded as an obstruction for modelling   iterated loop spaces in one step, even if there were a similar model theorem around (which -- to the knowledge of the author -- is not the case).