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3-manifolds efficiently bound 4-manifolds

2008; Wiley; Volume: 1; Issue: 3 Linguagem: Inglês

10.1112/jtopol/jtn017

1753-8424

Francesco Costantino, Dylan P. Thurston,

Advanced Graph Theory Research

It has been 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, but little attention has been paid to the complexity of the 4-manifold produced. Given a 3-manifold M3 of complexity n, we construct a 4-manifold bounded by M of complexity O(n2), where the ‘complexity’ of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation. 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 further prove that, for a manifold M satisfying the geometrization conjecture with Gromov norm G and shadow complexity S, we have c1G ⩽ S ⩽ c2G2, for suitable constants c1, c2. In particular, the manifolds with shadow complexity 0 are the graph manifolds. In addition, we give an O(n4) bound for the complexity of a spin 4-manifold bounding a given spin 3-manifold. We also show that every stable map from a 3-manifold M with Gromov norm G to ℝ2 has at least G/10 crossing singularities, and if M is hyperbolic there is a map with at most c3G2 crossing singularities.