3-manifolds efficiently bound 4-manifolds

Details 3-manifolds efficiently bound 4-manifolds 3-manifolds efficiently bound 4-manifolds

2005-06-28

arxiv.org/abs/math/0506577v3

Journal of Topology 1(3): 703-745, 2008

Mathematics

Geometric Topology

39 pages, 21 figures; added proof for spin case as well

Scientific paper

10.1112/jtopol/jtn017

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the complexity of the 4-manifold produced. Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the minimum number of n-simplices in a triangulation. It is an open question whether this quadratic bound can be replaced by a linear bound. The proof goes through the notion of "shadow complexity" of a 3-manifold M. A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M. We prove that, for a manifold M satisfying the Geometrization Conjecture with Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are the graph manifolds.

Affiliated with

Also associated with

No associations

Rating

3-manifolds efficiently bound 4-manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with 3-manifolds efficiently bound 4-manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and 3-manifolds efficiently bound 4-manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-434575