3-manifolds efficiently bound 4-manifolds Export

@misc{citeulike:239710, abstract = {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.}, archivePrefix = {arXiv}, author = {Costantino, Francesco and Thurston, Dylan P.}, citeulike-article-id = {239710}, citeulike-linkout-0 = {http://arxiv.org/abs/math.GT/0506577}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.GT/0506577}, day = {28}, eprint = {math.GT/0506577}, keywords = {3d, 4d, complexity, manifold}, month = {Jun}, posted-at = {2005-06-29 08:37:35}, priority = {2}, title = {3-manifolds efficiently bound 4-manifolds}, url = {http://arxiv.org/abs/math.GT/0506577}, year = {2005} }

TY - ELEC ID - citeulike:239710 L3 - citeulike-article-id:239710 N2 - 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. TI - 3-manifolds efficiently bound 4-manifolds KW - 3d KW - 4d KW - complexity KW - manifold AU - Costantino, Francesco AU - Thurston, Dylan PY - 2005/Jun/28/ UR - http://arxiv.org/abs/math.GT/0506577 ER -