3-manifolds efficiently bound 4-manifolds

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.

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

References in corpus (4)

• A calculus for branched spine of 3-manifolds
• Kashaev's invariant and the volume of a hyperbolic knot after Y. Yokota
• Simplifying triangulations of S^3
• The Smale Conjecture for lens spaces

Cited by in corpus (19)

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• On the Volume Conjecture for hyperbolic Dehn-filled $3$-manifolds along the figure-eight knot
• Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II
• Fundamental shadow links realized as links in $S^3$
• Shadows of acyclic 4-manifolds with sphere boundary
• Investigating additional structures on preimages of regular values of smooth maps induced from the manifolds via the Reeb spaces and applications
• Computation of leading coefficients in asymptotics of relative quantum invariants
• Fold maps and immersions from the viewpoint of cobordism
• An upper bound conjecture for the Yokota invariant
• Milnor fibration, A'Campo's divide and Turaev's shadow
• Pair of pants decomposition of 4-manifolds
• Stable maps and hyperbolic links
• On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs
• Colored Jones invariants of links in S^3 # k S^2 X S^1 and the Volume Conjecture
• Characterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of the Reeb spaces in some 3-dimensional manifolds
• Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-
• Shadows of 2-knots and complexity