In his talk at the meeting Agol sketched some of the ideas of the proof and outlined several applications to other problems in 3-dimensional topology. This was the topic of Canary's talk. For example, what is the Hausdorff dimension of its boundary?

## Handle decompositions of 3-manifolds

The other talks were chosen to cover as many different areas of topology as possible, and hence it is difficult to find an overall theme to describe them. Stefan Bauer described his recent work on invariants of 4-manifolds, including a refinement of the Seiberg-Witten invariants due to him and Furuta. In a more algebraic direction, Jesper Grodal described some of the latest developments in the field of 2-compact groups spaces which are complete at the prime 2, and whose loop space has finite mod 2 cohomology i.

The goal is to classify all simply connected 2-compact groups this has already been done at odd primes , and understand how close they are to being 2-completions of classifying spaces of compact connected Lie groups. In his talk, Grodal focused on the problem of defining root systems for 2-compact groups, and some of the problems which arise at the prime 2 and did not arise for odd primes.

In the field of geometric group theory, Martin Bridson talked about subgroups of direct products of hyperbolic groups, and described his counterexample with Grunewald to a conjecture of Grothendieck, where they construct a homomorphism of finitely presented, residually finite groups which induces an equivalence of representation categories but is not an isomorphism. When you do the surgery, you get a solid torus doubled along the boundary.

This is all written up very well in the book of Prasolov and Sossinsky. Proposition Figure 8.

Another very nice book where all this is explained is the book of Rolfsen. Here is how you can see this hat tip to Milnor.

## COVERINGS OF 3-MANIFOLDS BY THREE OPEN SOLID TORI

Here's a more 4-dimensional proof, or rather a 4-dimensional interpretation of Igor Rivin's answer above. Full disclosure: I've swept all issues about smoothing corners under the rug, but I hope that this at least gives some intuition. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 3 years, 8 months ago.

## Coverings of 3-manifolds by three open solid tori Pages 1 - 13 - Text Version | FlipHTML5

Active 1 year, 11 months ago. Viewed times.

Ivan Ivan 18 2 2 bronze badges. Igor Rivin Igor Rivin This amounts to the same thing.

### Your Answer

The complement is also a ball. Now add drill a hole through the middle of the ball. You get a solid torus. This has the effect of adding a solid tube to the exterior ball, making it a solid torus as well.