Building aspherical manifolds.

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 is a finite complex. Then there is a closed aspherical manifold and a retraction .

This manifold can be explictly constructed by gluing together copies of the regular neighorhood of 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 , which is not residually finite, and observe that the presentation 2-complex is aspherical, so we have a finite . Then using the theorem to "promote" this to a closed aspherical manifold, we get a manifold with fundamental group retracting onto . But a group retracting onto a non-residually-finite group is also non-residually finite, so we have found a closed aspherical manifold 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 .

Let be a simplicial complex, and , the vertices of .

From we construct two things: some complexes to glue together, and some groups with which to do the gluing. First, we construct the groups. Define to be the group , i.e., the abelian group generated by with . Next define to be the right-angled Coxeter group having as its Coxeter diagram; specifically, is the group with generators and relations for and also the relations if the edge is in . Note that is the abelianization of .

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

Having the complexes and the groups, we will glue together copies of along the ’s, thinking of the latter as the mirrors. Specifically, define with provided that and , where , and is the subgroup of generated by . That is a mouthful, but it really is just carefully taking a copy for each group element of and gluing along the ’s in the appropriate manner. The resulting compplex has a action with fundamental domain . Similarly, we use to define a complex .

The topology of is related to the complex that we started with. For example, if is the triangulation of , then is a manifold. Similarly, if is a flag complex, then is contractible.

The idea, now, is to take some finite complex , embed it in , and take a regular neighborhood; the result is a manifold with boundary , and with . Triangulate as a flag complex, and call the resulting complex . Instead of gluing together copies of , glue together copies of along the subdivision of to get and . With some work, we check that is contractible because is flag, and that the contractible space covers the closed manifold , which is therefore aspherical. Since is a retraction of spaces, we have found our desired aspherical manifold with a retraction of fundamental groups.