Building aspherical manifolds.

 January 25, 2007 talks mathematics

I gave a Farb student seminar talk on a lovely paper [1].

I also used some of the material in [2] 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 (\mathbb{Z}/2\mathbb{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 \mathbb{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.

[1] M.W. Davis, Groups generated by reflections and aspherical manifolds not covered by euclidean space, Ann. Of Math. (2). 117 (1983) 293–324.

[2] M.W. Davis, Exotic aspherical manifolds, in: Topology of High-Dimensional Manifolds, No. 1, 2 (Trieste, 2001), Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002: pp. 371–404.