Hyperbolization of Polyhedra

I gave a talk in the Farb and Friends Student Seminar (back in March!) on:

Davis, Michael W. and Januszkiewicz, Tadeusz. Hyperbolization of polyhedra. J. Differential Geom. 1991. 347–388. MR.

This is an awesome paper—well-worth a few words on every blog!

The construction is way easier than you might think. The ingredients:

• A model space $X$ with a map $f : X \to \Delta^n$
• Any simplicial complex $K$ with a nondegenerate (edge-non-collapsing) map $K \to \Delta^n$ (if having a map to $\Delta^n$ seems like a bother, note that the barycentric subdivision $K’$ comes with a map to $\Delta^n$ for free).

Let $X_J = f^{-1}(J)$ for $J$ a subcomplex of $\Delta^n$; we think of this as decomposing $X$ into pieces resembling a simplex.

Now the construction is easy: replace each simplex in $K$ with a corresponding piece of $X$. Or more formally, build the fiber product of $X$ and $|K|$ over $\Delta^n$; this fiber product is denoted by $X \tilde{\Delta} K$ in the paper. From this, we get a natural map $f_K : X \tilde{\Delta} K \to K$.

The vague upshot is this: features of $X$ translate into features of $X \tilde{\Delta} K$, while nonetheless preserving features of $K$. Here are a couple of examples of how assumptions on $X$ lead to consequence for $X \tilde{\Delta} K$.

• If $X$ is path-connected, and for each codimension 1 face $\alpha$ of $\Delta^n$, we have $X_{\alpha} \neq \varnothing$, then $\pi_1(f_K) : \pi_1(X \tilde{\Delta} K) \to \pi_1(K)$ is a surjection.
• If $X$ and $K$ are PL-manifolds, and $\dim X_J = \dim J$, and $\partial X_J = X_{\partial J}$, then $X \tilde{\Delta} K$ is a PL-manifold.

Building aspherical manifolds.

I gave a Farb student seminar talk on a lovely paper,

Davis, Michael W.. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2) 1983. 293–324. MR.

I also used some of the material in

Davis, Michael W.. Exotic aspherical manifolds. 2002. 371–404. MR.

which summarizes other the many applications of the “reflection group trick,” and works through some examples with cubical complexes.

The main result is

Theorem. Suppose $B\pi = K(\pi,1)$ is a finite complex. Then there is a closed aspherical manifold $M^n$ and a retraction $\pi_1(M) \to \pi$.

This manifold $M$ can be explictly constructed by gluing together copies of the regular neighorhood of $B\pi$ embedded in some Euclidean space. The application of this theorem is to “promote” a finite complex to a closed aspherical manifold. For instance, we have a finite complex with non-residually-finite fundamental group: define the group $\pi = \langle a, b : a b^2 a^{-1} = b^3 \rangle$, which is not residually finite, and observe that the presentation 2-complex is aspherical, so we have a finite $B\pi$. Then using the theorem to “promote” this to a closed aspherical manifold, we get a manifold $M^n$ with fundamental group retracting onto $\pi$. But a group retracting onto a non-residually-finite group is also non-residually finite, so we have found a closed aspherical manifold $M^n$ with non-residually-finite fundamental group.

Just to whet your appetite, let me introduce a few of the main players, so as to give a sense of how to glue together copies of the regular neighborhood of $B\pi$.

Let $L$ be a simplicial complex, and $V = L^{(0)}$, the vertices of $L$.

From $L$ we construct two things: some complexes to glue together, and some groups with which to do the gluing. First, we construct the groups. Define $J$ to be the group $(\Z/2\Z)^V$, i.e., the abelian group generated by $v \in V$ with $v^2 = 1$. Next define $W_L$ to be the right-angled Coxeter group having $L^{(1)}$ as its Coxeter diagram; specifically, $W_L$ is the group with generators $v \in V$ and relations $v^2 = 1$ for $v \in V$ and also the relations $v_i v_j = v_j v_i$ if the edge $(v_i,v_j)$ is in $L$. Note that $J$ is the abelianization of $W_L$.

Next we will build the complexes to be glued together with the above groups. Let $K$ be the cone on the barycentric subdivision of $L$, and define closed subspaces ${ K_v }_{v \in V}$ by setting $K_v$ to be the closed star of the vertex $v$ in the subdivision of $L$. Note that $K_v$ are subcomplexes of the boundary of $K$, and that a picture would be worth a thousand words right now.

Having the complexes and the groups, we will glue together copies of $K$ along the $K_v$’s, thinking of the latter as the mirrors. Specifically, define $P_L = (J \times K)/\sim$ with $(g,x) \sim (h,y)$ provided that $x = y$ and $g^{-1} h \in J_{\sigma(x)}$, where $\sigma(x) = { v \in V : x \in K_v }$, and $J_{\sigma(x)}$ is the subgroup of $J$ generated by $\sigma(x)$. That is a mouthful, but it really is just carefully taking a copy $K$ for each group element of $J$ and gluing along the $K_v$’s in the appropriate manner. The resulting compplex $P_L$ has a $J$ action with fundamental domain $K$. Similarly, we use $W_L$ to define a complex $\Sigma_L = (W_L \times K)/\sim$.

The topology of $\Sigma_L$ is related to the complex $L$ that we started with. For example, if $L$ is the triangulation of $S^{n-1}$, then $\Sigma_L$ is a manifold. Similarly, if $L$ is a flag complex, then $\Sigma_L$ is contractible.

The idea, now, is to take some finite complex $B\pi$, embed it in $\R^N$, and take a regular neighborhood; the result is a manifold $X$ with boundary $\partial X$, and with $\pi_1 X = \pi$. Triangulate $\partial X$ as a flag complex, and call the resulting complex $L$. Instead of gluing together copies of $K$, glue together copies of $X$ along the subdivision of $L$ to get $P_L(X) = (J \times X)/\sim$ and $\Sigma_L(X) = (W_L \times X)/\sim$. With some work, we check that $\Sigma_L(X)$ is contractible because $L$ is flag, and that the contractible space $\Sigma_L(X)$ covers the closed manifold $P_L(X)$, which is therefore aspherical. Since $P_L(X) \to X \to P_L(X)$ is a retraction of spaces, we have found our desired aspherical manifold $M = P_L(X)$ with a retraction of fundamental groups.

Approximating L^2 invariants by finite-dimensional analogues.

I gave a couple of seminar talks on

Lück, W.. Approximating $L\sp 2$-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 1994. 455–481. MR.

Here’s the main result in the paper. Let $X$ be a CW-complex, and filter $\Gamma = \pi_1 X$ as $\Gamma = \Gamma_1 \rhd \Gamma_2 \rhd \cdots$ with $[\Gamma_i : \Gamma_{i+1}] < \infty$ so that $\bigcap_i \Gamma_i = { 1 }$. Let $X_i$ be the cover of $X$ corresponding to the normal subgroup $\Gamma_i$.

Then, the limit of the “normalized” Betti numbers $\lim_{j \to \infty} b_j( X_i ) / [\Gamma : \Gamma_i]$ is equal to $b^{(2)}_j(X)$, the $L^2$ Betti number of $X$. In particular, the limit of the normalized Betti numbers is independent of the filtration! In other words, we have “approximated” the $L^2$ invariant by a limit of finite-dimensional approximations.

The awesome thing about this result is how “easy” the proof is; it’s just some linear algebra (eh, functional analysis), but I don’t claim to have a very conceptual understanding of why it is true. In the big book on this subject,

Lück, Wolfgang. $L\sp 2$-invariants: theory and applications to geometry and $K$-theory. 2002. xvi+595. MR.

there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

Algebraic topology and distributed computing.

I gave a seminar talk on

Herlihy, Maurice and Rajsbaum, Sergio. Algebraic topology and distributed computing–-a primer. 1995. 203–217. MR.

This paper doesn’t do it (but Rajsbaum’s MSRI talk did), but the result can be reformulated combinatorially, so that the algebraic topology appears as an instance of Sperner’s lemma; this is the sort of thing that should be done at mathcamp.

Here is something that amuses me, but I know that if anyone else said it, I would find it extraordinarily annoying: seeing as these results apply to anything (I mean, the local model of computation is irrelevent), this is an example of how deterministic systems, when combined with each other, yield non-deterministic results (though I have to be careful what I mean by “deterministic”—the system as a whole is determined, but non-deterministic from the perspective of the agents in that they cannot determine the outcome). Clearly I should write a philosophy paper, called “Free will and algebraic topology: a primer,” in which people are vertices in the simplicial complex of all possible worlds.

It will be better for all of us if I stop now.